It first clicked for me when reading Barr and Wells' Category Theory for Computing Science [0], but I don't know about your mathematical background. Category theory is algebra, so it's probably advisable to study basic group theory before tackling category theory. (I have a hard time seeing how functors make sense until you've understood the general concept of a homomorphism, which is perhaps easiest to do in the context of groups).
[0]: http://www.case.edu/artsci/math/wells/pub/ctcs.html