This is a question coming from somebody who has had at most some high school physics. It would help if in your profile you added your age and/or field.
From the level of the question I will assume the above.
Explanations I have read of why photons are emitted from atoms mention electrons being 'excited' to another energy level, and then returning to their base level, releasing a photon.
This comes about because the microcosm is quantized, i.e. obeys the rules of quantum mechanics, and at a first level of explanation,the nucleus of the atom is a potential well into which the electron, when speaking of hydrogen, can have only quantized specific orbits. The orbits are defined as energy levels. If, for some reason, the electron is at a higher energy level and there exists a level below, it falls to that level releasing a photon which has energy h*nu equal to the difference in energies of the two orbits. Usually the excitation comes about from a kick to the lowest energy electron of an appropriate energy to send it to a higher energy orbit, or to ionize the atom. For atoms with a higher atomic number, the energy levels are filled up sequentially with electrons which again can behave as above.
I have also seen the occasional mention of 'fields' and I vaguely expect from what I've read that interaction of the electrons with these fields has some effect.
In the above, first level picture, the field is the electromagnetic field, and mainly the effective field coming from the charges of the nucleus and the orbiting electrons.
I've also seen, but not used, Feynmann diagrams that involve additional particles.
Feynman diagrams come into the next level of complexity of the theory of elementary particles. The description above pertains to first quantization. Feynman diagrams belong to second quantization and are a powerful and necessary tool for calculating processes in elementary particle physics, like electron electron scattering etc. This needs a course in Quantum Field Theory.