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.