Why is QED renormalizable?

QED has only a finite number of irreducible divergent diagrams. The main notion of divergence of a diagram is power-counting: The term every diagram represents has the form of a fraction like $$ frac{intmathrm{d}^n p_1dotsintmathrm{d}^n p_m}{p_1^{i_1}dots p_k^{i_k}}$$ and you can compute the difference between the momentum power in the numerator and denominator and call it $D$. Heuristically the diagram diverges like $Lambda^D$ in a momentum scale $Lambda$ if $D > 0$, like $ln(Lambda)$ if $D=0$, and is finite if $D < 0$. This can fail - the diagram can be divergent for $D < 0$ - if it contains a smaller divergent subdiagram.

If you work out the general structure of $D$ for the diagrams of QED, you should be able to convince yourself that QED has only a finite number of divergent one-particle irreducible diagrams. That cancelling the irreducible diagrams is enough to cancel iteratively the divergences in all higher-order diagrams containing them in arbitrary combinations to all orders is a non-trivial statement sometimes called the BPHZ theorem, whose technical meaning - though not by this name - is explained by the Scholarpedia article on BPHZ renormalization.