I will refer to the Axiom of choice as ($AC$). As far as I know, the widely accepted axioms of set theory is the Zermelo-Fraenkel axioms together with the $AC$. But the last axiom seems to be the most special out of these axioms.
I have read that the $AC$ is the most controversial axiom in the entire history of mathematics. Yet it remains a crucial assumption not only in set theory but equally in modern algebra, analysis, mathematical logic, and topology (I think when we use the Zorn´s lemma, we also use the properties of the $AC$?).
How I understand the $AC$ intuitively is that it is needed when you need to make infinitely many arbitrary choices at once.
My question is: For many other axioms, we really can tell "yes, this property makes sense to demand." We know from the real world. But is this the case of $AC$?
If not, probably somebody in the history decided it is important and that we need to demand $AC$. That´s why I want to ask:
What is the origin of the axiom of choice, why was it introduced?
What are areas of math where it helps? Was there any particular problem that needed the $AC$ to be solved? And only after that it was discovered to be useful also in other areas?
Thank you for response and if needed, I can try to make any part of the question more specific.