This proof sketch of the second isomorphism theorem uses the symbols in my writeup there without repetition of the formulation, so you may want to read that first.

Define a homomorphism f: G -> G/H by f(g) = gH. Then f is clearly an epimorphism. Since N is a subgroup of H, it's easy to see that the induced map f: G/N -> G/H defined by f(gN) = gH is a homomorphism. Now note that Ker f is precisely H/N, and that f is an epimorphism. The theorem follows from applying the first isomorphism theorem to f.