Feynman diagrams are used to compute correlation functions perturbitively. A correlation function is the probably of observing a specific quantum state at later time "t" when starting from a specific quantum state at earlier time "t". The way you do this is compute <ψ|(time propagation)|ψ>. Naively, we just use the time propagation operator exp(-i*integration(H, dt)*t), but when we expand in a power series, we need to time-order the H's for which you use Wick's theorem. Then the correlation function for each term in the time-ordered expansion is pictorially represented by a Feynman diagram.