Proofs in second-order logic work the exact same way as proofs in first-order logic: You prove a sentence by beginning with axioms, and construct the sentence using inference rules. In first-order logic, you have individuals, functions (from individuals to individuals), terms, and predicates (loosely speaking, functions from individuals to truth values), but may only quantify over individuals. In second-order logic, you may also quantify over predicates and functions. In third-order logic, you add functions of functions of individuals, and, in fourth-order logic, the ability to quantify over them. This continues all the way up to omega-eth order logic (a.k.a.: higher-order logic), where you may use functions of arbitrary orders. The axioms and inference rules are the same as in first-order logic.

Wait, I said nothing about sets above. Sets are no problem: a set containing certain elements is equivalent to a predicate which only returns true on those elements.

I also lied about it being the same as first-order logic -- a couple are added. One very useful axiom scheme is comprehension, which roughly means that you can find a predicate equivalent to any formula. You can think of this as being an axiom schema of first-order logic, except that, since it cannot be stated until fourth-order logic, it contains no axioms in first-order logic.

(My formal logic training is primarily in Church's theory of types, which is very similar to higher-order logic [though superficially extremely different]. I probably mixed-up terminology somewhere in the above.)