Show that any monotone Boolean function is computable by a circuit containing only AND and OR gates
Any Boolean function can be written as a DNF. Each clause in the DNF specifies one truth assignment for which the function holds. For example, the DNF form of XOR is $(x \land \lnot y) \lor (\lnot x \land y)$.

The main observation is that if the function is monotone, you can remove all the negated literals (why?). Once you do that, you get a formula for the function which uses only AND and OR. Some clauses in this formula subsume others (for example, $x$ subsumes $x \land y$). If you drop all the subsumed clauses, you arrive at the minterm representation of the function. (A minterm is a minimal satisfying assignment.)

It's a bit more confusing, but you can also do everything using CNFs. This way you arrive at the maxterm representation.