Generally speaking, nobody claims that the axioms of set theory are true in any ordinary sense of the word true. Does it makes sense in the real world to assert the existence of an infinite set, and have different sizes of infinities as a consequence? It is irrelevant, you assert the existence of an infinite set because it is practical from a pure mathematical point of view, because you want to model infinite sets like the natural numbers (an abstraction, do they exist? it doesn't matter to mathematicians generally speaking).
Mathematics no longer pretends to describe what is true, what holds in our reality. That idea was abandoned some time ago. A canonical example were non-euclidean geometries, that were studied in the XIXth century for the sake of it. They had applications later, but the motivation for their study and changing Euclid's axioms was formal.