I don't get the Axiom of Choice. All it seems to say is that if you have a bunch of piles of objects, it is possible to pick exactly one item from each pile. Well big fucking whoop, what's the big deal? Why is this so controversial?
It's not controversial. Famous would be a more accurate word. It's one of those things, like Euclid's parallel axiom, that was just sort of assumed for a good amount of time because it was so "obvious" that no one bothered to inspect it. Then someone did, and guess what? They couldn't prove it. Furthermore, a number of well known results in set theory, analysis, and algebra require the axiom of choice (or a weaker variant of it).