What are large cardinals for?
Solution 1:
I will leave the explaining of large cardinals to someone more knowleadgeable and explain one place where they are useful in tidying up things: category theory. In category theory, you are constantly faced with proper classes (the category of all sets, of all groups, etc.). To make things worse, you want to form functor categories, but due to the sizes of the classes involved there is no way to do that in ZFC (and other set theories like NBG or MK soon hit a wall of their own). The Grothendieck axiom of universes is a large cardinal axiom (and I am told that it is rather mild compared to the large cardinals that set theorists routinely consider) that allows us to tidy up things for this and other constructions without paying attention to set sizes. Truth be told, this is mainly a convenience, as if one paid proper attention, and at the cost of circumlocutions (and vast tribulations) one could avoid them. But really, why go to all this trouble to settle what is a minor technicality not germane to the problem at hand, when you have this labor-saving device at hand?
Solution 2:
G. Rodrigues's specific answer gets at the general issue: large cardinals are used to examine how much more one can proof in ZFC set theory. The first time I discovered large cardinals (in Jech's 2000 book Set Theory), I was amazed. A large cardinal is just a "very big" set, after all, but I did not realize that the existence of such a set changed the nature of what was mathematically provable. For example, there is, according to Jech, the event that started it all: Ulam's work on the problem of measure. It is well-known that Lebesgue measure over the reals is not defined for all sets, but it turns out to be undecidable in ZFC alone if any non-trivial measure on the reals exists at all. In order to get such a measure, one must assume the existence of a large cardinal, which is now called a measurable cardinal. So I think of large cardinals as things that change the very nature of the mathematical "plumbing". Deep stuff.
On a more practical level, I think it was Dudley who said that large cardinals can be useful for seeing why a proof is failing: if a proof is not working, seeing if it fails at a large cardinal can provide insight.