There are different ways of approaching axiomatization. One is more "top down". You have a pretty good idea what the truths are about a particular subject matter, and the problem is to find some reasonably managable set of principles from which those truths all follow. Axiomatizations of logic itself might be so construed. It's arguable that the fundamental notion here is really validity, the semantic notion, defined in terms of interpretations and the like, and then the problem is to find a set of axioms from which all the valid formulae will be derivable. Whether the axioms have some intuitive basis may be neither here nor there.

Of course, that needn't be the only way of looking at the matter, and it doens't seem terribly plausible in the case of set theory, especially after one's naivete has been shattered by the paradoxes. Here more of a "bottom up" perspective might seem appropriate: One might think that the axioms of set theory ought in fact to have some kind of intuitive basis. And, as it happens, there are ways of presenting axiomatizations of set theory that make the intuitive basis of the axioms of ZF rather more apparent than it might be in standard presentations. One nice guide to this is George Boolos's paper "The Iterative Conception of Set".

Boolos actually argues in a later paper, "Iteration Again", that not all of the axioms of ZFC can be justified in this fashion. It does not, of course, follow that they cannot all be justified (though Boolos explores that view, too, in another paper, "Must We Believe In Set Theory?"). An alternative suggestion is that some of these axioms---for example, the axiom of replacement---are justified only `pragmatically', that is, because they get us what we want. So, on this view, the axioms of ZFC would have a kind of mixed origin: some in our "intuitive" grasp of the notion of set, some in our sense of what the truths about sets ought to be.