Hi, Scott. Forget about Weinberg's proof for a while and picture this line of thought. Relativistic QFT comes nicely by generalizing single particle relativistic quantum mechanics which in turn generalizing the old quantum mechanics of Schroedinger, Dirac and Heisenberg. The latter is of course a mind-blowing extension of the classical mechanics of Hamilton. What can you say about an observable O in Hamilton mechanics ? It's defined on the phase space (here O becomes a function of p and q) by the property that, when evaluated on the surface of the solutions of Hamilton's equation, it's a mere numerical constant.
Going in reverse, p and q become operators in quantum mechanics, no longer variables of the phase space, so do the quantum mechanical observables become functions of the 'fundamental' operators p, q. But p and q in a single Hilbert space (actually in a RHS, but that's a finesse you won't need) can be linked to a and a^dagger. So the observables become functions of a and a^dagger. And now you go from a single HS to a Fock space and voila', you find Weinberg's statement.
