Proper classes come up when you exhaust the means of forming sets. You need a set when you need to know the means of set theory have not been exhausted -- for example when you want to go on and form a colimit of the structures you have formed so far. Exactly when the means are exhausted, depends on what means of forming sets you have.

First take an example that exhausts second order arithmetic but does not exhaust Zermelo set theory (or simple type theory): the etale fundamental group of an arithmetic scheme. There is no universal cover like the ones for topological spaces and this is not a logical or set theoretic problem but inherent in the situation. (The scheme has etale covers of any finite degree, so a universal cover could have no finite degree.) So Grothendieck and others formed the colimit of all symmetries of the (non-universal, actually existing) etale covers. Second order arithmetic suffices to give the symmetry group of any one etale cover, but because we want the colimit of all these, we need an uncountable group. Second order arithmetic will not produce that. Third order will.

Grothendieck and Dieudonne often found they wanted colimits sort of like this, over all cases of some structure, but not just all that exist in second order arithmetic. Naively put, they wanted all that exist in set theory. Maybe all algebras over some ring, or all finitely generated algebras. They knew there is a big difference between those examples, since there is not even a set of all algebras over a ring up to isomorphism (in any set theory they considered). Choosing one countably infinite set of generators will give you a set of all finitely generated algebras over that ring up to isomorphism. But in either case they did not want to bother with such details. And they were all the more eager to avoid analogous details in more complex cases.

If you really want to talk about all sets, or all natural weak equivalences of functors from Top to Top, or all generalized normal subobjects of $S^2$ in the homotopy category then you are exhausting the means of set theory (though the last two cases are less obvious than the first).

Grothendieck and Dieudonne appreciated the point perfectly. They knew workarounds to fit some of their larger constructions into ordinary set theory, and they were confident other workarounds could be found. But they were not interested in that. They saw that when they used all sets etc., it was not "all" in any metaphysical sense. It was all those constructed by the ordinary means of set theory, so they posited one non-ordinary means of constructing sets: each set is contained in a universe. At any point they work inside some universe, so what would be proper classes in ordinary set theoretic accounts are sets in the next larger universe.