Every student of set theory knows that the early axiomatization of the theory
had to deal with spectacular paradoxes such as Russel's, Burali-Forti's etc.
This is why the (self-contradictory) unlimited abstraction axiom ($\lbrace x | \phi(x) \rbrace$
is a set for any formula $\phi$) was replaced by the limited abstraction
axiom ($\lbrace x \in y | \phi(x) \rbrace$ is a set for any formula $\phi$ and any
set $y$).
Now this always struck me as being guesswork ("if this axiom system does not work,
let us just toy with it until we get something that looks consistent "). Besides, it is not
the only way to counter those "set theory paradoxes" -there's also
Neumann-Bernays-Godel classes.
So my (admittedly vague) question is : is there a way to explain e.g. Russel's paradox
that does better than just saying, "if you change the axioms this paradox disappears ?" Clearly, I'm looking for an intuitive heuristic, not a technical exact answer.
EDIT June 19 : as pointed out in several answers, the view expressed above is historically false and unfair to the early axiomatizers of ZFC. The main point is that ZFC can be motivated independently from the paradoxes, and "might have been put forth even if naive set theory had been consistent" as explained in the reference by George Boolos provided in one of the answers.
Best Answer
George Boolos has a number of very readable (to the non-expert like me) essays on this subject. Try "The Iterative Conception of Set" in Logic, Logic and Logic. He tries to find a way to look at the axioms of ZF set theory from a perspective that makes them look natural and not simply contrived to avoid paradoxes. I don't know anything about how Boolos's views are seen by other Set Theorists.