How does a photon released from particle A interacting with particle B, entangle them?

First of all, the photon needs to be entangled with the particle that emitted it. That is not difficult to accomplish. One can for instance use the conservation of momentum to obtain an entangled state: $$ |psirangle = sum_n |mathbf{p}_nrangle_{phot} |mathbf{p}-mathbf{p}_nrangle _Aalpha_n , $$ where $sum_n |alpha_n|^2 = 1$ and $mathbf{p}$ is the combined momentum of both particle A and the photon. When the photon now interacts with particle B, it will transfer an amount of momentum that is correlated with that of particle A. Hence, the two particles become entangled.