The story of Quantum Field Theory(QFT) treats particles such as electron,proton or photon very differently from quantum mechanics(QM) and classical mechanics(CM).
We all know with CM we can't explain the behavior of an atom because according to Maxwell's theory accelerating particles should radiate their energy and for an atom this means that electrons should fall into nucleus by radiating energy.
QM says that electrons have defined stationary orbits and manages to explain phenomenon like black-body radiation. But QM also fails when particles have speed comparable to speed of light.
Now QFT is born which is a marriage of QM and Special Theory of Relativity. QFT treats particles as excited states of a field and interaction between particles is studied by interaction between fields. Standard Model is a result of QFT. If we want to study interaction between two electrons, we say that one electron applies force on other by virtue of photons. QFT also says that in vacuum a photon can create a electron-positron pair and electron-positron pair can annihilate to produce a photon. Of course the principal of conservation of energy and principal of conservation of momentum should be respected in any kind of process.
To explain the phenomenon of photon emission or photon absorption when electrons changes their orbits, the picture of QM works pretty well because electrons never reach speed of light unless they are put in an accelerator. Explaining the phenomenon by QFT, we can describe electron by an excited state of field 1(Dirac Field) and similarly photon by excited state of field 2(Maxwell Field). Now if photon is absorbed then the field 2 comes to it's ground states and field 1 moves to upper excited state. The mathematics of this process is not very easy and moreover it is very lengthy. If you want to see the mathematics I will strongly recommend chapter 8 for Dirac Field, chapter 9 for Maxwell Field from book Ashok Das-Lectures on QFT. For interaction of these two fields you can refer to chapter 5 from book Peskin,Schroeder- Introduction to QFT.