Basically, because a lot of mathematical proofs involve phrases like "Let X be the set of all things with property Y", and contain a tacit assumption that the "set of all things with property Y" is in fact a legitimate set that obeys the standard rules of set theory (which the proofs then go on to invoke).

If you start with "let X be the set containing all sets that do not contain themselves", and accept that such a set exists, you can prove anything, and while that particular phrasing might make the pitfall obvious, it's not as obvious what other rules might lead to paradox.