What "forces" us to accept large cardinal axioms? Large cardinal axioms are not provable using usual mathematical tools (developed in $\text{ZFC}$). 
Their non-existence is consistent with axioms of usual mathematics.
It is provable that some of them don't exist at all.
They show many unusual strange properties.
$\vdots$
These are a part of arguments which could be used against large cardinal axioms, but many set theorists not only believe in the existence of large cardinals, but also refute every statement like $V=L$ which is contradictory to their existence.
What makes large cardinal axioms reasonable enough to add them to set of axioms of usual mathematics? Is there any particular mathematical or philosophical reason which forces/convinces us to accept large cardinal axioms? Is there any fundamental axiom which is philosophically reasonable and implies the necessity of adding large cardinals to mathematics? Is it Reflection Principle which informally says "all properties of the universe $V$ should be reflected in a level of von Neumann's hierarchy" and so because within $\text{ZF-Inf}$ the universe $V$ is infinite we should add large cardinal $\omega$ (which is inaccessible from finite numbers) by accepting large cardinal axiom $\text{Inf}$ and because $V$ is a model of (second order) $\text{ZFC}$ we should accept existence of inaccessible cardinals to reflect this property to $V_{\kappa}$ for $\kappa$ inaccessible and so on? 
Question. I am searching for useful mathematical, philosophical,... references which investigate around possible answers of above questions. 
 A: Let me give an argument, not that we should believe large cardinal axioms or their consistency, but rather that regardless of our belief in consistency we should still be interested in results around them.
First, large cardinals are uniquely useful in analyzing principles of strong consistency strength. That is, we know from experience that there are many principles, with consistency strength greater than $ZFC$, for which large cardinals function as a useful organizational principle. This is especially important if I'm agnostic about the consistency of large cardinals (which I am), since then I'm also agnostic about a bunch of other fairly natural principles and want a nice "yardstick" to organize my knowledge of them.
Even if I actively believe, say, that "there is an inaccessible" is inconsistent with $ZFC$, playing with large cardinals is still useful to me: if I believe inaccessibles are inconsistent with $ZFC$, then I also must believe that "DC + every set of reals is Lebesgue measurable" is inconsistent with $ZF$; the point is, there are reasonably natural philosophical viewpoints which reject inaccessibles - say, believing that the Inner Model Hypothesis is true - which yield, via arguments around large cardinals, philosophical positions against other principles for which no such natural viewpoint is known to exist.
A: The line of reasoning you mention at the end of your post, firmly in support of large cardinals, was first argued forcefully in 


*

*W. N. Reinhardt, “Remarks on reflection principles, large cardinals, and elementary embeddings,” Proceedings of Symposia in Pure Mathematics, Vol 13, Part II, 1974, pp. 189-205 


and the ideas are further discussed, explained and basically supported in 


*

*Penelope Maddy, Believing the axioms. I, J. Symbolic Logic 53 (1988), no. 2, 481--511.

*Penelope Maddy, Believing the axioms. II, J. Symbolic Logic 53 (1988), no. 3, 736--764.
These articles have now a rather large literature of discussion and criticism in the philosophy of set theory. To get started, you might find further resources on the reading list of my recent course NYU Philosophy of Set Theory. One can now find numerous articles arguing on any given side of each issue. 
A: There are (possibly) two questions here:


*

*Why should we believe that large cardinal axioms are consistent?

*Why, if we believe that large cardinal axioms are consistent, should we believe that they are true?
Here are some reasons for believing that large cardinal axioms are consistent (or at least, that small large cardinal axioms that have been studied for a long time are consistent.)
First, there is the empirical fact no one has published a proof of a contradiction from the assumption $\mathsf{ZFC} + {}$"there is an inaccessible cardinal" (for example) despite a long period of study in which many theorems have been proved from this assumption.  Although some large cardinal hypotheses (such as Reinhardt cardinals) have turned out to be inconsistent, this was discovered relatively quickly, in the period during which most people were still skeptical of them.
Second, there is "fine structure" which gives canonical models for the smaller large cardinal axioms (so far, up to Woodin cardinals and a bit further.)  It seems reasonable to expect that a systematic study of the structure of the models of a theory would eventually reveal the inconsistency of the theory if it were inconsistent, and this has not happened yet.
For question 2, let us now assume (informally, for the sake of non-mathematical argument) that large cardinal axioms are consistent.  Why should we then believe that they are true?
Most people find it natural to believe the assumptions that they use in their day-to-day work, so this question is closely related to the question of which axioms people should use.  Of course, the answer will depend on what types of theorems they want to prove.  In most areas of mathematical research, $\mathsf{ZFC}$ seems to be sufficient in a practical sense and there does not seem (to me) to be a compelling argument that people working in these areas should use, or believe, any kind of axiom beyond $\mathsf{ZFC}$.
So perhaps the question should be revised to "why should mathematicians who want to prove theorems beyond $\mathsf{ZFC}$ use large cardinal axioms, instead of alternatives such as $V=L$?"  A practical answer is that doing so allows us to prove lots of interesting theorems.  Suppose that I assume $\mathsf{ZFC} + {}$"there is a measurable cardinal" and you assume "$\mathsf{ZFC} + V=L$."  Then for every theorem that you prove, I could have proved (if I were clever enough) a corresponding theorem of the form $L \models \ldots.$  On the other hand, I may have the opportunity to discover an interesting theorem about measurable cardinals that you do not have the opportunity to discover (unless you investigate countable transitive models with measurable cardinals, which seems like an unnatural thing to do if you believe that $V=L$, even though it is presumably formally consistent for you to assume the existence of such models.)
This last point is summarized by the slogan "maximize interpretive power."  Many of the points I made above are better made in the following paper.  I think that what I wrote here leans toward Steel's viewpoint, but I do not claim to have rendered it faithfully.

Feferman, Solomon; Friedman, Harvey M.; Maddy, Penelope & Steel, John R. (2000). Does mathematics need new axioms? Bulletin of Symbolic Logic 6 (4):401-446.

A: This is a personal opinion rather than an answer (in fact another personal opinion of mine is that this kind of question cannot have a meaningful objective answer).
Compare this situation with Euclidean geometry. It is not quite correct to ask whether one should believe in the fifth postulate or not. This is because with the current state of knowledge there is no problem at all to deal with all possible versions of it. And in fact, already situations when the status of the fifth postulate varies from point to point are very well understood.
In set theory also, already state of knowledge is ripe to study if not all, then at least significant amount of possibilities which can arise from various combinations of large cardinal (and several other important) axioms. And in fact it is perfectly meaningful to consider and study mathematical structures which allow for variability of the status of these axioms similarly to the variation of curvature on a geometric surface.
I believe that in such circumstances the question of belief becomes obsolete. It is true that in physics one may believe that the universe is positively or negatively curved, or flat. But this is because we are placed inside this universe. In case of mathematics, we are not placed inside any particular model of set theory, hence we are not forced to choose. Certainly some models are distinguished among the rest by some special properties, like flat geometry is distinguished among the rest of geometries, but that's all one can say I think.
A: It is useful in the category theory, in particular, in its applications to algebraic geometry. "Small" categories (whose objects form a set) are much nicer than "large categories" (whose objects mere form a "class", whatever it means). In algebraic geometry one wants to consider categories like Sets, Schemes and so on as small --- and technically it is done using Grothendieck's "Axiom of universe", which is equivalent to existence of strongly inacessible cardinals large than a given cardinal, see http://en.wikipedia.org/wiki/Grothendieck_universe
A: This is only a partial answer, but Harvey Friedman has a research program to find concrete $\Pi^0_1$ sentences that are purely combinatorial (i.e., make no reference to concepts from logic such as axioms or formal systems) that can be deduced from a large cardinal axiom and that imply the consistency of a (slightly weaker) large cardinal axiom.  The $\Pi^0_1$ statement can of course be partially verified by direct computation, so if you convince yourself that it is true, then the large cardinal axiom helps "explain" why it is true.  I believe that Friedman has carried out his program up to and including subtle cardinals; see this post on the Foundations of Mathematics mailing list, for example.
I believe Friedman is optimistic that his program can in principle be carried out for any large cardinal axiom, but at present I believe he has no natural, explicit $\Pi^0_1$ statements that require (say) measurable cardinals to prove.
A: Reflection principles are a good starting point, but applying them, well... Although worldly cardinals are the "smallest type" of ZFC-unprovable cardinals, let's jump up to the uncountable inaccessibles. The classical argument is that V is (strongly) inaccessible, i.e. it is not the successor of any number, it is cofinal with itself, and is not the powerset of any smaller set (indeed, V itself is not quite a set at all). Therefore there is a limit set that is cofinal with itself and not the powerset of any smaller set. So far so good. But notice that 0 and $\aleph_0$ are equal to their own cofinalities, and are not powersets of smaller sets, and can be construed as limits. So don't they satisfy the reflection principle, here? You might object, "But yeah, we're talking about uncountable inaccessibles." So you would have V be uncountable, too. However, suppose you adopted ZFC sans replacement. In that event, $\aleph_{\omega}$ counts as a large cardinal (its existence is unprovable in ZFC sans replacement). More importantly, then, it is a shadow of V in that compromised instance of ZFC. But since $cf(\aleph_{\omega}) = \aleph_0$), we are in a position where V is not quite inaccessible, after all, and in fact is not even perfectly uncountable as such, either. At any rate, the V-reflection argument for, "There are inaccessibles," must be amended to, "There are uncountable inaccessibles," but at this point the justification of that assertion has been much weakened.
There's a deeper problem with reflection principles, though, which can be highlighted by formulating them in Ackermann's terms. According to Ackermann, reflection principles are grounded in the ultimate principle, "Absolute infinity is unknowable." However, knowledge is justified true belief (plus some Gettier-dimensional quality). So are we building our theory of set-theoretic justification on a concept of what is not justified? That seems wrong, to me.
Now Gödel spoke of intrinsic and extrinsic justification, here. The former roughly translates Kant's doctrine of analytic knowledge. Analytic knowledge is grounded in an unfolding of a concept's content. So Gödel said that if we unfold the iterative concept of set, we will have justified (true!) beliefs about that concept. In the era of set-theoretic multiverses and logical pluralism, on the other hand, we can talk about unfolding all manner of conceptual contents: as long as the concept we are referring to has an elementhood relation encoded into it, we will be analyzing a concept of sets. Anyway, Penelope Maddy refined Gödel by having us iterate isomorphism types: these are what we are MAXIMIZING as we approach absolute infinity. (She claimed, IIRC, that hyperset axioms don't give us new isomorphism types, so that was why nonwell-founded set theory wasn't justified by MAXIMIZE as such. However, it is difficult for me to believe that, "Hyperset A is isomorphic to hyperset B," is a false kind of sentence, or that the isomorphism, here, would be absolutely identical to some well-founded kind. So I think Maddy's reasoning is deficient, in this case.) Large-cardinal axioms involve these unfolding isomorphism types, ergo...
On the other hand, Maddy's doctrine of intrinsic justification only seems to justify assertions like, "There is at least one uncountable inaccessible." But we know that we can contrive set theories in which there are any number of uncountable inaccessibles: finitely many (for any n), transfinitely many (for any $\kappa$), or class-many. Maddy's doctrine is of qualitative, not quantitative, iteration: we have already quantitatively MAXIMIZED, after all, as soon as we have admitted any proper class of sets whatsoever. So though Maddy's doctrine looks like a good way to ground assertions that various types of large cardinals exist, it is not clear (to me) that she provides us with any solid way to settle questions like, "How many cardinals are there, that map from those types?"1
The arguably dominant method of set-theoretic ontology (nowadays) is, however, to advert to generic consistency, and hold that consistent sets of ontological sentences are justified by the illustration of their consistency. This is a kind of "plenitudinous Platonism." (With the advent of paraconsistent set theory, granted, we open the doors to a plenitudinous Platonism that does not depend on illustrations of ontological consistency. But that is an aside...) The prevailing case of this doctrine is reference to a set-theoretic multiverse, where the sets of justifiable ontological sentences are converted into the various universes. Inasmuch as Kant originally identified analytic knowledge with a kind of knowledge based on the laws of identity and noncontradiction, we seem to have in the consistency-theoretic method of ontology, the perfect crystallization of the notion of "intrinsic justification" (as analytic).2
All that being said, there is a different way to apply the concept of intrinsic justification, to the issue at hand. And this is to unfold the content, not of the iterative (or any other such) concept of sets, but of the concept of justification itself. Granted, the furthest along that we've gone here, in the mainstream, is justification logic. But it turns out that we can very easily and strongly justify a number of large-cardinal axioms, in light of justification logic as such.

Case 1. Have there be an axiomatic set theory involving justification logic simpliciter; call this "jZFC." For every such theory T, there is a relatively worldly cardinal, i.e. at least one $\kappa$ such that $V_{\kappa}$ is a model of T. But jZFC is intrinsically justified. Accordingly, the axiom of a worldly cardinal for jZFC inherits this intrinsic  justification. (Reasoning: model theory itself justifies various mathematical assertions. So a set theory of model-theoretic justification, inherits the justificatory force of model theory.) Therefore, the axiom, "There is at least one jZFC-worldly cardinal," is as justified as can be.


Case 2. In proof-theoretic ordinal analysis, there are some proof-theoretic ordinals that are to be marked out by using "ordinal collapsing functions." These functions take other large countable ordinals (exceeding the proof-theoretic threshold) for their inputs. In turn, these large countable ordinals can be characterized as mirrors of various genuinely large cardinals. So if jZFC has a proof-theoretic ordinal assigned to it (I see no reason to assume otherwise), then we can reason upwards from that to a jZFC-relevant large countable ordinal, and from there to a genuine large jZFC-relevant cardinal. I have next to no idea what other properties such a cardinal would have; but the assertion that such a cardinal exists, seems well-justified by the reasoning at hand.


Case 3. Use an infinitary justification logic to background jZFC. There are characterizations of large cardinals that can be designed by reference to matters of normal infinitary logic, e.g. weakly or strongly compact cardinals. So there should be such large-cardinal characterizations available in terms of jZFC. Again, these axioms, as expressions of the concept of justification itself, seem to be intrinsically justified, very strongly, as such. Also, since (for example) strongly compact cardinals "oversee" measurable ones, the jZFC-relevant strongly compact cardinals then ground assertions of measurable cardinals in general, and some bracketing of the, "How many?" question, there, in particular.

I have an idea for justifying axioms of Reinhardt sets, too, in this context, but it depends on disengaging too much with established set theory, to explain as of now. I'll just leave you with the possibility that jZFC could be invoked to justify3 asserting the existence of Reinhardt sets, then.
1Admittedly, one could probably make up some sort of MAXIMIZE-dimensional reason to move from, "The various types of large cardinals exist," to, "There are class-many of each type." Also, some large $\kappa$ are such that they encode for a lower type of large cardinals, to the effect that we can infer the existence of $\kappa$-many cardinals of the lower type, given below $\kappa$. E.g., since the least measurable $\kappa$ is strongly inaccessible, we get $\kappa$-many strongly inaccessibles below the least measurable.
2On the flip side: if the Kant-Frege thesis that "existence is not a (first-order) predicate" is true, that seems to rule out this talk of justifying an existence claim using the consistency-theoretic method. This is because analytic justification is, therefore, never existential justification as such: all existence claims are synthetic. Personally, though, I wonder whether we might say: it is amiss to speak of existence being analytically true of other things, but is it amiss to speak of other things being analytically true of existence? Or what of an assertion such as, "Existence exists"? I suspect, that is, that we could justify large-cardinal axioms as analytic of the concept of existence: they are forms of existence, that themselves exist. (How are they to be such forms? But remember how Conway describes the meaning of $2^{\aleph_0}$: every real number is individuated by countably many subindexes. In other words, the greater and greater infinite sets, allow us to individuate more and more kinds of otherwise finite numbers. If (as Quine said) to be is to be the value of a bound variable, then the further and further ways of having values and bound variables, are further and further ways "for" things to exist. (Or: if the logic of existence is the logic of existential quantification...) Presto: the higher and higher cardinals are manifestations of higher and higher forms of existence itself, hence can be taken for analytic truths of pure existence. (Going a little sideways, we can use this idea about analytic ontology to "explain" the ontological argument: it is not that existence is analytically true of God, but that God, representative of a form of existence (the created/uncreated dichotomy), is therefore analytically true of existence. (Hence ironically, though Aquinas disavowed the ontological argument, his own claim that God is somehow "subsistent being itself" is tantamount to a recapitulation of that argument, after all.))
3I think we have to differentiate between what jZFC can prove, and what it can justify. By definition, essentially, jZFC cannot for example prove that there is a relatively worldly cardinal. However, it does seem as if it can justify this claim. Now, in justification logic, there are all kinds of second-order questions that come up; modulo jZFC, we have questions like, "Is the assertion, 'X has been proved,' itself justified? And can I prove assertions like, 'X is justified'?" But I don't really know the answers to these questions.
