Why do we need model categories? I cannot give a good answer to this question. And
2) Why this definition of model category is the right way to give a philosophy of homotopy theory? Why didn't we use any other definition?
3) Has model category been used substantially in any area not related to algebraic topology? 
 A: Let me try to turn this question around in the following way: we need model categories since we do not really understand why such a perfectly symmetric structure can capture the very essence of homotopy-related phenomena.
When I first saw axioms of a model category I was at the same time awestruck by breathtaking beauty of the Quillen's masterpiece, and puzzled by the obvious fact that this seemingly purely aesthetic conceptual structure is very efficient at describing lots of specific situations where we have the notion of homotopy in various very general senses.
Let me mention one particular aspect of model categories that has been, and still remains, especially baffling for me. In most typical examples, cofibrations are, at least "morally", monomorphisms, while fibrations tend to be epimorphisms. But in the closely resembling structure of factorization system it is exactly the opposite - the left halves of the factorization are presumed to "behave like" epis and the right ones like monos.
There are several other mysteries related to model categories. Let me mention just one more. Along with Grothendieck's derivators, there are several closely related structures developed by Franke, Heller and several others. They suggest that a model category is actually the tip of an iceberg, encoding a flood of structures derived from it. I agree with the opinion that from modern viewpoint thus it is more natural to switch to $\infty$-categories; but is it yet well understood how a model category structure encodes all these higher order structures so efficiently?
So my answer is - we need model categories to understand what they are trying to tell us about homotopy theory.
A: Extensive answers have already been given in this thread. Just a few remarks here and there.


*

*I think the question “why we need” assumes something about “we”, and in some extent, about “need”. There are people who study those objects for their own sake, there are those who study applied PDE and have no need of model categories. Even those who enter the domain which can be covered by model categories often work in a specialised situation where more adapted techniques exist (homological algebra and triangulated categories for algebraic geometry being an example). I guess the meta-reason is that at the current historic moment (or maybe that will remain so indefinitely) any description of entities ‘’up to homotopy’’ from scratch cannot be done without having this ‘’up to homotopy’’ notion already defined. One has to break the vicious circle and pull things down from their ‘’platonic’’ world. Which brings me to the answer for 

*Since categorical philosophy has proven itself to be reasonably natural, introducing categories with weak equivalences is both a natural step and something reflected in many examples. As we now know, this principle of modelling homotopy phenomena is, formally speaking, just as good as any other higher-categorical approach. Working with an arbitrary category with weak equivalences $(C,W)$ can be impossible in practice, but there are many ways to (homotopically) embed it into a model category.  One can consider the category of simplicial presheaves on $C$ for instance, with a suitable Bousfield localisation of the projective model structure. (The examples usually present themselves with better, often canonical, embeddings.) One can thus work with the objects of $C$ by performing operations on the bigger model category and verifying that the answer is sensible for $C$. This adds to the explanation as why the notion is quite ubiquitous. 

*Algebraic geometry (both the field and the community) has different tradition from algebraic topology, yet model categories have found their way here as well. For example, Kontsevich defines a noncommutative space as a suitable DG-category, and considers them up to Morita equivalence. There are a few model structures on DG-categories, which cover both the usual DG- and Morita equivalences, and they have been used to get various results, such as the theorem representing DG-functors between quasicoherent sheaf categories as bimodules. Another issue is the structure of noncommutative cohomological invariants, Hochschild cohomology, Deligne conjecture, and related matters. Many of the results (including laying a foundation of derived algebraic geometry) were obtained by people outside of the core algebraic topology community, with the use of model categories.

A: Model categories capture the idea that in many cases you resolve an object by an equivalent object that is better behaved. The standard example is replacing a chain complex by a chain complex of projectives (or injectives), which is quasi-isomorphic. This is used both in building the derived categories and in deriving functors. Model categories are used almost everywhere, where similar ideas are important. Examples: 
1) One can resolve commutative rings by simplicial free commutative rings. Deriving the module of differentials leads to Andre-Quillen homology. 
2) We replace a space by a weakly equivalent CW-complex. 
3) We resolve an operad (e.g. in chain complexes) by a better behaved "cofibrant" operad. For example, we replace the associative operad by an $A_\infty$-operad. 
Examples (1) and (3) are of great importance also outside of algebraic topology. Other examples come from applying homotopical thinking to non-homotopical situations. For example, in motivic homotopy theory one considers model structures on simplicial presheaves on the category of smooth schemes over a fixed base $S$ to build the homotopy theory of motivic spaces. In some sense, one is resolving here schemes by simplicial presheaves. 
One can certainly say that model categories are a very powerful framework important both in and out of algebraic topology. This does not mean that they are the right way to do homotopy theory for every task at hand. For example, sometimes we do not need to speak about fibration and cofibrations at the same time and one can use the theory of cofibration categories, which is a little less powerful, but also more flexible. Sometimes we do not want to focus on (co)fibrations at all (in some sense they are just a tool to understand the weak equivalences better). Then we use $(\infty,1)$-categories instead (e.g. in the guise of quasicategories developed into a full-fledged theory by Joyal and Lurie). They are especially useful for structural reasoning (for example, they allow us to take a homotopy limit of a diagram of homotopy theories, which is not really possible in the language of model categories). 
A: This answer is an elaboration on Dylan's comments.
1) Let us define a homotopy theory to be a pair $(C, W)$, where $C$ is a category and $W$ is some class of morphisms called weak equivalences.
(Let's say that $W$ satisfies some reasonable properties, e.g. it contains isomorphisms and is closed under composition.)
Of course, the archetypal example is given by topological spaces with the class of weak homotopy equivalences, which are continuous maps that induce isomorphisms on homotopy groups.  This is the homotopy theory that has been studied classically by algebraic topologists.  Similarly we may consider the category of simplicial sets, again with the class of weak homotopy equivalences.  Another classical example of a different flavour is given by the category of chain complexes of $R$-modules, for a commutative ring $R$, together with the class of quasi-isomorphisms (morphisms which induce isomorphisms on homology groups).  This homotopy theory is also known as homological algebra.
2) Now, we would somehow like to express the idea that the homotopy theories of topological spaces and simplicial sets are equivalent, even though the categories themselves are far from being equivalent.
That is, we would like to equip the category of homotopy theories itself with a class of weak equivalences (I'm going to ignore size issues here).
One idea is to consider Gabriel-Zisman localization: given any homotopy theory $(C, W)$, there is a canonical construction $C[W^{-1}]$ which universally inverts all morphisms in $W$.
Thus we could say that an equivalence of homotopy theories is an equivalence of the associated Gabriel-Zisman localizations.
The homotopy theories of topological spaces and simplicial sets will then be equivalent in this sense, according to a theorem of Milnor.
The downside of this definition is that any homotopy theory $(C,W)$ will be indistinguishable from the homotopy theory $(C[W^{-1}], isos)$ (where we take weak equivalences to be isomorphisms).
Experience has taught us that is not what we want: the construction $C[W^{-1}]$ is poorly behaved from the homotopical point of view, as it is not even possible for instance to recover homotopy (co)limits from $C[W^{-1}]$, while ideally we would expect that any homotopy theory should have an internal notion of homotopy (co)limits.
Instead, one should use a much more refined version of the construction $C[W^{-1}]$, namely the $(\infty,1)$-categorical localization; it can be modelled for example by the Dwyer-Kan localization of simplicially enriched categories.  This gives a good notion of weak equivalence of homotopy theories.  We then have a striking theorem of Clark Barwick and Dan Kan that can be paraphrased as follows:

The homotopy theory of homotopy theories is equivalent to the homotopy theory of $(\infty,1)$-categories (as modelled by quasi-categories,
  complete Segal spaces, simplicially enriched categories, etc.).

We can then view the pair $(C,W)$ as a presentation or model of the associated $(\infty,1)$-category.  For example, the homotopy theories of topological spaces and simplicial sets are both models of the same $(\infty,1)$-category.
3) The theorem of Barwick-Kan tells us that, from the perspective of homotopy theory, there is no difference between homotopy theories or say, quasi-categories.
However, from the perspective of category theory, these two models are very different.
The question is about how to access categorical information in a given homotopy theory $(C,W)$.
That is, in ordinary category theory, we are used to talking about objects, morphisms, functors, limits and colimits, presheaves, the Yoneda lemma, and so on.
We have $(\infty,1)$-categorical versions of all these things, but how do we see them inside a given pair $(C, W)$?
Of course, we may always look at the respective operations in the underlying category $C$, but these will in general not be compatible with our class of weak equivalences (e.g. the (co)limit of two weakly equivalent diagrams may not be weakly equivalent).
We can view the theory of model categories as a solution to this problem: the idea is to endow the pair $(C,W)$ with a model structure, i.e. the structure of cofibrations and fibrations in an appropriately nice way.
Doing this is usually nontrivial, but when possible, it gives us a powerful way to compute things like homotopy (co)limits, namely by computing them in the underlying category $C$ after taking suitable (co)fibrant replacements.
We should keep in mind though that what we really care about are homotopy theories.  Despite the effectiveness of model categories, it is a fact that the choice of specific model-categorical presentation adds a factor of arbitrariness to all constructions and proofs, and as a result does not always allow us to express ourselves quite as fluently as we are used to in ordinary category theory.  Nowadays it is also common to work with other models of $(\infty,1)$-categories instead of working directly with homotopy theories; for example, the model of quasi-categories, developed by Joyal and Lurie, has an amazingly well-behaved category theory (even if it comes with its own difficulties).  At the end of the day, these are all just equally valid approaches to working with homotopy theories.
A: Today we understand that what we are really interested in when we talk about "homotopy theory" are in the end "$\infty$-categories".
In fact I have even heard some peoples claim that maybe in the future model categories would not be needed anymore and we will only talk about $\infty$-categories (in this post $\infty$ means $(\infty,1)$) . Even if I do not completely agree with that, and I think a majority of people will not either (and anyway only the future will tell) it is definitely a point of view following which the answer to your question 1 and 2 are "we don't" and "mostly because of History".
But even from this "extreme" perspective model category are interesting because they are the simplest way we have at our disposal to construct and study locally presentable $\infty$-categories: it is a lot simpler to construct a model category and to compute things with it than to construct and work with, for example a locally presentable quasi-category.
So this is why we need model categories: it is just simpler to handle than any other model we have to represent these objects (the locally presentable $(\infty,1)$-category). And nowdays $\infty$-categories are becoming hugely important in various area of mathematics (but that would be for a follow up question "why do we need higher categories").
Regarding your second question:
Well It is definitely a good definition, that cover 95% (or more) of the case that we would like to cover and it is convenient to work in. But it is definitely not the only definition that can play this role. There is a lot of other (very similar but generally weaker) notion that could replace model categories and that we need to cover these last "5%". The one that arise the most often in pratice are the "Left semi-model structure", and a little less often the "right semi-model structures", which are both weakening of the notion of model categories that still allows to do essentially everything we do with model categories and that cover at least 4.5% on these last 5%. Then after that you have a all jungle of weaker notion (see for example Cisinski's paper on derivable categories). In my opinion the main reason to prefer model categories to these weaker notions is that there is a huge amount of results proved for model categories and you never know how they will extend to the weaker cases. So I would say that for your question (2) my answer would be "mostly because of history, but with very good reasons to do so and with no clearly better alternative known"
For your question (3) one can actually argue that "no" if you have model category then you are related to algebraic topology. The main point here is the following: as soon as you have a model category you have "functions spaces" and a weak actions of the homotopy category of space on your model category. (essentially: the homotopy category of space plays the role that the category of sets has for ordinary category).
So as soon as you have model categories appearing in some area of mathematics it actually mean that this area is connected to algebraic topology. The best example of this is type theory: it has been studied for years by category theorist, computer scientist and logician, but at some point Voevodsky (and partial result in this direction also existed a little bit before) realised one could interpret it in certain model category, and the area became "homotopy type theory" and is definitely connected to algebraic topology.
This being said there are some examples of model categories that are not totally related to usual algebraic topology. Very famously you have the topic of Motivic homotopy theory which studies various model structure in algebraic geometry, or you have a model category for the homotopy theory of $C^*$-algebra, or here a model category on a certain type of model of intentional type theory, or the obvious folk model structure on the category of categories which just model the notion of equivalences of categories.
You also have a whole bunch of model structures in the theory of higher category theory that are more and more distant from ordinary algebraic topology (so it depends on how large is your definition of "algebraic topology").
A: Model categories provide a powerful framework for commuting (homotopy) limits and colimits,
and, more generally, for commuting left adjoint functors and (homotopy) limits,
as well as right adjoint functors and (homotopy) colimits.
(In what follows, I omit the adjective “homotopy” before limits and colimits.)
In finitely presentable ∞-categories filtered colimits commute with finite limits
and in finitely presentable ∞-categories with a set of compact projective generators
sifted colimits commute with finite products.
However, many other situations of interest are not covered by such statements.
For instance, one might want to commute a sifted colimit past an infinite product, a pullback,
or a cosifted limit (e.g., a cosimplicial totalization).
One might also want to commute sifted colimits past finite products
in ∞-categories that do not have a set of compact projective generators,
e.g., the ∞-category of small ∞-categories.
In all such situations the relevant statement is false at least for some diagrams,
so whatever criterion we devise must analyze the specific diagrams at question.
This is precisely what model categories achieve.
For example, one might want to commute K-indexed colimits (for some small diagram K)
past some right adjoint functor F: C→B.
(For instance, one can take F to be lim_L: Fun(L,B)→B, the limit functor for L-indexed diagrams,
which will allows us to commute K-indexed colimits past L-indexed limits.)
We would like to devise a condition on a diagram D: K→C that would guarantee that the canonical
comparison map colim_K F(D) → F(colim_K D) is an equivalence in the ∞-category B.
This is achieved by the following creative procedure.
First, consider the relative ∞-category (i.e., ∞-category equipped with a class of maps (weak equivalences) closed under composition)
Fun(K,C) whose weak equivalences are created by the functor colim_K (i.e., a natural transformation of functors K→C
is a weak equivalence if its K-colimit is an equivalence in the ∞-category C).
The ∞-category Fun(K,B) is turned into a relative ∞-category in the same way.
The functor Fun(K,F): Fun(K,C)→Fun(K,B) is a functor between ∞-categories that need not preserve weak equivalences.
Now comes the (potentially) creative part: equip the relative ∞-categories Fun(K,C) and Fun(K,B)
with model structures such that Fun(K,F) is a right Quillen functor.
In many situations of interest this can be done immediately using existing tools.
Our criterion now says that if D: K→C is a fibrant diagram,
then the comparison map colim_K F(D) → F(colim_K D) is an equivalence in the ∞-category B,
i.e., the colimit of D over K commutes with F.
Different choices of model structures give us different criteria.
For instance, one can take K=Δ^op, B=spaces (alias ∞-groupoids),
C=Fun(L,B) (i.e., L-indexed diagrams of spaces)
and F=lim_L: Fun(L,B)→B, the L-indexed limit functor,
where L can be taken to be the pullback diagram or the infinite discrete category
and the model structure can be taken to be the projective model structure.
In this case we obtain a criterion for commuting
Δ^op-indexed colimits past L-indexed limits.
This recovers, for example, the traditional methods
for computing homotopy pullbacks of simplicial sets (replace one of the legs by a fibration),
infinite homotopy products of simplicial sets (fibrantly replace all terms), etc.
There are many variations on the above theme, for instance,
one can consider weighted limits (in the sense of enriched category theory)
and then derive the limit functor with respect to both the functor and the weight,
which yields even more powerful computational tools etc.
Many classical results on model categories fit in the above framework.
For instance, if M is a simplicial model category,
X is a cofibrant object,
and Y is a fibrant object,
then the simplicial mapping space Map(X,Y) from X to Y
computes the mapping space in the ∞-localization of M with respect to its weak equivalences.
We can fit this in the above framework by taking F=Map(-,Y): Fun(Δ^op,M)→Fun(Δ^op,Spaces),
and observing that a cofibrant object X has a cofibrant cosimplicial resolution X⊗Δ^
To answer the original questions:
1) We need model categories
because we need to commute homotopy colimits and limits,
more generally, left adjoints and homotopy limits, or right adjoints and homotopy colimits.
Such commutation statements are unique to the formalism
of model categories and cannot be obtained in other formalisms
(without reconstructing a substantial part of the theory of model categories).
Another way of saying this is that we want to model operations on objects of interest to us
using their “presentations” (alias “resolutions”), which typically
are (co)limit diagrams whose homotopy (co)limit is the given object.
Model categories explain how operations on resolutions model operations on objects themselves.
2) Relative categories work just as well for setting up abstract theory, and in fact all formalisms
for ∞-categories (e.g., quasicategories, relative categories, complete Segal spaces, etc.)
are the same for all practical purposes once the basic theory is set up.
(The fact that one hardly ever writes down a quasicategory
explicitly is a testament to this claim.)
However, in all of the above cases one cannot directly work with the underlying platonic notion of ∞-category (alias (∞,1)-category)
and is forced instead to use some presentation (i.e., model, hence the name “model category”).
This presentation is nothing else than a colimit diagram of some shape (e.g., a simplicial diagram
in the case of Joyal's quasicategories).
The desire to manipulate such presentations (i.e., colimit diagrams) efficiently leads us straight to model categories (and the above setup).
This explains why all foundational work on these formalisms by Joyal-Lurie, Barwick-Kan, Rezk, etc. uses model categories.
3) Yes, e.g., in functional analysis (see the work of Costello-Gwilliam, for example).
In quantum field theory one has to perform homological algebra with infinite-dimensional vector spaces of smooth functions.
Model categories are bound to show up either explicitly or implicitly (for Costello-Gwilliam mostly the latter so far, but this may change soon).
A: Topologists had been studying homotopy theory long before we conceived of homotopical algebra; the definition of model category is abstracting how they did so.
The modern philosophy is that homotopical algebra is $\infty$-category theory (which is short for $(\infty, 1)$-category). In practice, this is often done by reducing questions of $\infty$-category theory to questions of 1-category theory (i.e. the usual category theory).
A particularly common form of doing this is to sandwich an $\infty$ category $\mathcal{C}$ between two 1-categories $C$ and $h\mathcal{C}$ with functors
$$ C \xrightarrow{L} \mathcal{C} \xrightarrow{\pi} h\mathcal{C}$$
such that this arrangement has the following properties. Let $W \subseteq C$ be the subcategory of arrows $w$ for which $\pi L(w)$ is an isomorphism.


*

*If $w$ is an arrow of $C$, then $w \in W$ if and only if $L(w)$ is an equivalence.

*If $D$ is a 1-category and $F : \mathcal{C} \to D$ is a functor, then there is a unique1 functor $G : h\mathcal{C} \to D$ such that $F \simeq G \pi$.

*If $\mathcal{D}$ is an $\infty$-category and $F : C \to \mathcal{D}$ is a functor such $F(w)$ is an equivalence for every $w \in W$, then there is a unique1 functor $G : \mathcal{C} \to \mathcal{D}$ such that $F = G L$.


1: up to unique1 equivalence
In short, $\pi : \mathcal{C} \to h\mathcal{C}$ is universal among arrows from $\mathcal{C}$ to 1-categories, and $L : C \to \mathcal{C}$ expresses $\mathcal{C}$ identifies $\mathcal{C}$ as the localization of $C$ obtained by turning the arrows of $W$ into equivalences.
The pair $(C, W)$ is called a saturated relative category, or a saturated homotopical category. The category $h\mathcal{C}$ is called the homotopy category of $\mathcal{C}$.
So, the functor $C \to h \mathcal{C}$ gives an avatar in 1-category theory for the $\infty$-category $\mathcal{C}$. Furthermore:


*

*We can identify which such functors are avatars of $\infty$-categories: they're the ones for which $h\mathcal{C}$ is the localization of $C$ by making everything in $W$ an isomorphism

*Every $\infty$ category can be expressed in this manner


and so this gives a way to work with $\infty$-categories in general, entirely in the settings of 1-category theory. Note that one usually specifies the pair $(C, W)$ rather than the functor $C \to h \mathcal{C}$. (and often weakens the condition on $W$)
However, we're greedy: we don't want to just have avatars of the categories, we also want to reduce everything we can to 1-category theory.
Model categories are extra structure that lets us do a lot of that. The fibrations and cofibrations satisfying the weak factorization axioms give us algebraic tools for reducing questions about $\mathcal{C}$ to the corresponding question about $C$.
For example, if you have a diagram $F : J \to C$ where every vertex is fibrant and every arrow is a fibration, then $L(\lim F) \simeq \lim(LF)$: that is, you can compute the limit in $\mathcal{C}$ simply by computing the limit in $C$. By "fibrant replacement", you can reduce any diagram $J \to \mathcal{C}$ to one of the above form, and so this gives us a way to compute limits in $\mathcal{C}$ by computing limits in $C$. (at least, when the index category is a $1$-category)
A: Voevodsky won a Fields Medal for resolving the Milnor Conjecture in number theory. His work fundamentally used model categories, and kicked off the field of motivic homotopy theory. Model categories have also been used a lot to answer questions in representation theory. If you google "stable module category" and "cotorsion pairs" you'll find lots of papers. Another example is the work of Finnur Larussen in complex analytic geometry (using model categories), or the work of Joachim and Johnson on Kasparov K-theory. There are applications of model categories to dynamical systems and computer science - look into work of Gaucher, of Bubenik, of Sanjeevi Krishnan, of Kathryn Hess, and of the people these papers cite. And of course, there's the recent work of Hill-Hopkins-Ravenel resolving the Kervaire Invariant One problem from differential topology using (the model category of) equivariant spectra. There are so many examples of the uses of model categories outside of algebraic topology, and of places where model structures were needed.
Other answers in the last few minutes have focused a lot on (2), and I've also written a lot of expository material on that (as have MANY others over the years). So, I focused in the paragraph above on your questions (1) and (3). Generally, though, I think this is too broad for mathoverflow, and duplicates Akhil's question that the first comment linked you to.
