Another perspective on the Axiom of Choice would be to formulate it as “you can select one element from each set in a given set of sets, and the selected elemen

Another perspective on the Axiom of Choice would be to formulate it as “you can select one element from each set in a given set of sets, and the selected elements form a set”. So we accept the existence of infinite sets (since we have an axiom stating that infinite sets exist), but acknowledge that they are tricky and that not every “collection” of whatever constitutes a set. For example, a collection of all sets does not constitute a set. In this interpretation, you can still choose one element from a set of sets, but the Axiom of Choice tells you in addition that the result of such choosing is not just a collection of elements, but itself a set, so you can apply all other axioms and theorems of set theory to it.