Comment by krick
Somehow the existing answers don't satisfy me, so here's my attempt. The essence of it is really simple.
The axiom is an obviously true statement: if you have a bag of beans, you can somehow take one bean out of it, without specifying, how do you choose the exact bean. Obvious, right? And that's really it, informally this is the axiom of choice: we are stating that we can somehow always do that, even if there are infinitely many beans and infinitely many bags, and the result of your work may be a collection of infinitely many beans.
Now, what's the "problem"? If you look closer, what I've just said is equivalent to saying we can well-order[0] any set of elements, which must make you uncomfortable: you may be ok with the idea that in principle you can order infinitely many particles of sand (after all, there are just ℕ of them), but how the fuck do you order water (assuming it's like ℝ — there are no molecules and you can divide every drop infinitely many times)?
This is both why we have it — ℝ seems like a useful concept so far; and the source of all notorious "paradoxes" related to it — if you can somehow order water, you may as well be able to reorder details of a sphere in a way to construct 2 spheres of the same size.