Interpretation of $W(J)$ in Quantum Field Theory

In Zee’s Quantum Field Theory in a nutshell, he discusses the quantity $W(J)$, where

$$left<0middle| e^{-iHT} middle|0right> = e^{-iET} = Ce^{iW(J)}$$

and for a free theory with a source $J = J_1 + J_2$ where $J_1$ and $J_2$ are localized in space, we have

$$W(J) = -frac{1}{2}int frac{d^4k}{(2pi)^4}J_2^*(k) frac{1}{k^2-m^2+ivarepsilon}J_1(k)$$

From here he says that there is a resonance peak at $k^2 = m^2$, the energy-momentum relationship for a particle and says that we interpret the physics as the following:

"In region 1 in spacetime there exists a source that sends out a ‘disturbance in the field,’ which is later absorbed by a sink in region 2 in spacetime."

I do not understand how you can come up with the explanation. Where we get that there is a disturbance/a particle that moves and is created and absorbed?