To my understanding, most mathematical theories can be simply understood in the view point of Category theory and its derivative theories. But what exactly is the motivation to study infinity category theory?
Why should one, for example, look at morphisms between morphisms and so on?
There are many motivations, but the short answer is that many desirable properties are only available in the world of $\infty$-categories. This is a wonderful miracle.
This is particularly visible when one works with objects up to a notion of equivalence that is coarser than equality or isomorphism. For instance. If we want to work with spaces up to homotopy or with chain complexes up to quasi-isomorphism. The main interest of category theory, when we want more from it than producing a convenient language to express the concept of natural transformation, is that it produces very powerful constructions, namely adjunctions and Kan extensions (they appear for instance in the formation of operations on sheaves, on representations of groups). The problem is that such constructions are known to exist systematically only if we work with categories having sufficiently many (co)limits, whereas the operation of enforcing non-trivial maps to become actual isomorphisms typically destroys the property of having enough (co)limits. This is why classically, we have to perform category theoretic constructions in rigid settings (topological spaces, chain complexes...) and see how they are compatible with our favorite notion of equivalence, which can become tedious.
In contrast, in $\infty$-category, all ordinary categories fit in. In fact, we can do all the usual constructions and theorems of ordinary category hold. For instance, if I consider the category of topological spaces and want to work up to homotopy, I can invert homotopy equivalences formally, exactly as in ordinary category theory (i.e. through a universal property that looks the same at first glance). Except that in the context of $\infty$-categories, this process does not produce an ordinary category any more, but a genuine $\infty$-category, and the latter has all small limits and colimits. Similarly if I want to invert formally quasi-isomorphisms of chain complexes (or, more generally, weak equivalences in any model category structure). That means that we can apply the traditional methods of category theory to our theory of spaces up to homotopy, without going back to the classical category of spaces and wondering if what we are doing is compatible with homotopy equivalences. But the price to pay is that we must work with genuine $\infty$-categories. And then, while we know how to express ordinary mathematics within $\infty$-category theory, we discover new phenomena that have no occurrence in classical mathematics; for instance, there are stable $\infty$-categories, i.e. those in which it is amazingly easy to produce a short exact sequence: $\infty$-categories with finite (co)limits in which the initial and terminal objects coincide and in which a commutative square is a pushout iff it is a pullback (exercise: if this is an ordinary category, it must be equivalent to the category with a unique object and a unique morphism, namely the identity). Such $\infty$-categories do not only exist but are the core of homological algebra (including in the traditional sense) and can be found everywhere. This leads to a more general ambidexterity questions that are extremely fruitful in stable homotopy theory and in representation theory, for instance; but this this simply the foundation of all homological algebra. Another example: one can do geometry in this language, and many moduli spaces get better regularity properties in the higher setting than their truncated counterparts.
At the end of the day, $\infty$-category theory looks very much like ordinary category theory, except that we can always reduce our computations to contexts in which there is only one way to identify objects: isomorphisms. This has to be compared with the zoo: equality, isomorphism, equivalence of categories, equivalence of 2-categories, homotopy equivalence, quasi-isomorphisms... That turns the process of gluing mathematical objects much more natural (in fact possible) in $\infty$-category theory, which is the basic tool to do any kind of geometry. That is why there is no turning back, I think.
"Why should one, for example, look at morphisms and so on?"
In some sense people have been looking at morphisms between morphisms as long as they've been looking at morphisms. At least, they've been looking at natural maps (which are maps between functors, i.e. maps between morphisms in the category of categories) as long as they've been looking at categories and functors.
It is said that when Eilenberg and Mac Lane invented category theory they needed categories and functors in order to make sense of what they meant by natural maps.
Let me try to give an algebraist's answer. I struggled with exactly this question for many months in the context of the stable module category, and it took a good many conversations with some of the best practitioners, to arrive at an answer that satisfied me.
My context is as follows. Let $G$ be a finite group and $k$ a field of characteristic $p$ dividing $|G|$. Then the stable module category $\mathop{\rm stmod}(kG)$ of finitely generated $kG$-modules is a tensor triangulated category. The large stable module category $\mathop{\rm StMod}(kG)$ consists of all $kG$-modules, and is hard to reconstruct from the small one, because filtered colimits require you to lift to the module category first. If you don't do this, you lose the "phantom maps" - namely the maps that factor through a projective module on restriction to every finitely generated submodule.
Now, if you regard $\mathop{\rm stmod}(kG)$ not as a triangulated category but as a stable infinity category, then there is enough extra structure that you can recover $\mathop{\rm StMod}(kG)$ as the ind-completion. The phantom maps somehow come for free.
On the other hand, you can get away with quite a bit less. If you regard $\mathop{\rm stmod}(kG)$ as the homotopy category of Tate resolutions of finitely generated modules, in other words as the homotopy category of exact complexes of finitely generated injective $=$ projective $kG$-modules, then there is a natural enhancement to a differential graded category, and as a differential graded category, ind-completion still makes sense. And it turns out that the extra information in the infinity category is essentially the same as that in the differential graded category. So for example equivalences at the level of infinity categories are exactly the same as equivalences at the level of differential graded categories. In particular, these are the "stable equivalences of Morita type" in modular representation theory.
But there is at least one twist to the story. One problem with the differential graded version is that the tensor identity, which is the Tate resolution of the field of coefficients, is only a tensor identity up to "all higher homotopies" and not strictly. The infinity category version, on the other hand, manages to have a "strict enough" tensor identity to get around any problems this might cause.
I may still have some misconceptions about this, and I hope that if I do, some kind soul will correct me without making me feel too bad about it. Toby? Anyone?
The need for $n$-categories arises naturally from considerations of $(n-1)$-categories, so before going into the $n=\infty$ case I think it pays off to think about some important examples of $2$-categories.
First of all, categories themselves form a $2$-category, because functors have natural transformations between them. Most category-theoretic constructions only work properly when categories are treated up to equivalence rather than up to isomorphism, and formalizing equivalence requires this $2$-categorical structure. The concept of adjunction is also inherently $2$-categorical.
Another interesting example is the $2$-category of bimodules. It has rings as objects, $(A,B)$-bimodules as morphisms between rings $A$ and $B$, with tensor product as composition. It is a $2$-category because $(A,B)$-bimodules themselves form a category rather than a set, and the tensor product is only associative up to isomorphism rather than "on the nose". The equivalence between rings viewed as objects in this $2$-category is precisely Morita equivalence, so this $2$-category is the natural language for Morita theory.
This should really be a comment on Alexander's answer, but comment size limitations and all that.
To my mind, 'infinity category theory' means fully weak infinity categories, something not really studied yet but still naturally motiavted by considerations like those mentioned in Alexander's answer.
We are naturally thrust into the realm of $1$-categories when we want to study structure on sets, and further thrust into the realm of $2$-categories when we want to study structure on categories; more generally, we are foisted into the realm of $n+1$-categories when we want to study structure on $n$-categories for any fixed $n<\omega$.
What if we want to study structure on all of these things at once? And bam, welcome to infinity category theory land -- no need for '$\omega-1$-categories' as suggested by lambda if we want motivation to study infinity categories, just $n$-categories for all $n$ and universal quantification.
Of course, this implicitly means that 'true' infinity category theory is really a study of how structure works in full generality; this is a desirable thing IMO, independent of any other considerations.
