More exactly, the nth term in the sum (n going from 0 to infinity) is, ignoring a combinatorial factor, the product of n copies of the interaction Hamiltonian density, each evaluated at a different space-time point, and arranged so that earlier events always come first, integrated over all space time (n times, for each point). Because of the nice algebraic properties of Hilbert spaces, if we want to evaluate the expectation of S between two states, we can just get the expectation of the product of Hamiltonians between those states, integrate over space-time, and sum over n.