Why does this need to be stated explicitly? Why does the axiom of choice not enjoy the privilege of being a first-class axiom like other axioms? Why do many proofs need to state the assumption of this axiom explicitly before using it?
Moreover, this axiom is _independent_ of the other axioms in ZFC. It is in fact possible to have entirely self-consistent "worlds" of mathematics, ones where axiom of choice is true, and ones where it is false.
More details and examples of alternate axioms are in the Wikipedia article: https://en.wikipedia.org/wiki/Axiom_of_choice
If it seems weird that math can give you contradictory results, remember that the difference only shows up when you deal with some form of infinity (e.g. when performing an operation on an infinitely large set). For any usage of math in the real world, the truth or falsity of this axiom won't give you contradictory results.
- Collection of non-empty sets with empty cartesian product.
- Infinite set without a countable infinite subset.
- There is a pair of sets such that neither is equinumerous with a subset of the other.
More or less, our intuition about seemingly obvious ideas is completely thrown off without choice. Banach-Tarski at least has the property that it probably doesn't directly apply to the real world (if you can split an object into probably physically impossible sets and then rejoin them correctly then you can double the volume of the physical object) and so doesn't really violate our intuition -- the premise doesn't apply in the real world, so no conclusion really matters. It's like claiming that every element of the empty set is a leprechaun with a pot of gold -- it's true, but it doesn't matter in any meaningful sense.