Is there a intuitive way to understand "the existence of a basis for every vector space is equivalent to the Axiom of Choice"?
Well, the proof itself goes through some other equivalents as well. So it might be slightly easier to understand the full circle:
$(1)\implies(2)$ is a well-known argument of transfinite induction, and $(2)\implies(3)$ is the classical use of Zorn's lemma. $(3)\implies(4)$ can be intuitively comes from the fact that linear combinations are finite, so we can define a vector space and from its basis choose the finite subsets of the family. Finally, $(4)\implies(1)$ can be handwavingly described as reiterating the subset process until we are left with just singletons from which the choice is canonical.
The proofs themselves of course require a lot more details, and the above intuitive and handwaving explanation is far from sufficient to fully get them. But that's the main intuition, I think.