By what mechanism is a photon emitted or absorbed in atomic electron state transitions?

Neither the photon nor the electron are classical particles and there is no Newtonian picture of the process. Instead you have to imagine relativistic fields that describe the probabilities to detect photons and electrons in different spacetime points. Before the absorption there is a non-zero probability to detect the photon and the probabilities of the electron field are roughly those predicted by the Schroedinger equation for the low energy state of the atom. After the transition the probability to detect the photon is mostly gone and the electron distribution is now in a higher state.

One has to be very careful even with this picture, since one can't do continuous measurements on this system without disturbing it. What these distributions really mean is that we prepare one photon, then perform one measurement on either the photon or the electron. We repeat this experiment many times and then we plot the probability distribution as a function of time. This would have to be a multidimensional plot because of the ways the parts of the quantum system interact. Unfortunately people are not very good at recognizing the finer details of such multi-dimensional phenomena. Whenever we talk about these probability distributions and we show images in books, the problem has already been greatly simplified for our own convenience. From the perspective of human perception it is probably next to impossible to visualize the entire process without some simplification or loss of information.