Why do electrons emit photons?

 Electrons carry electric charge. Electric charge is the source of the electromagnetic field. So electrons interact with the electromagnetic field. In a quantum field theory, this interaction between the electronic field and the electromagnetic field comes in set chunks, set units at any given frequency/energy. Therefore, whenever an electron interacts with the electromagnetic field, this interaction is in the form of emitting or absorbing such a unit, or quantum, of electromagnetic field energy. That quantum is known as the photon.

In contrast, electron neutrinos do not interact with (emit or absorb) photons at all, despite the fact that apart from their lack of electric charge and smaller mass, they are just like electrons. On the other hand, W- bosons, which are, very crudely speaking, just like electrons with their electron-ness removed (that is, an electron can emit an electron neutrino and turn into a W- boson), do interact with (emit and absorb) photons.

So it really is the electric charge. If there was no electric charge, the electromagnetic field would exist just by itself, without interacting with anything else. No photons would be emitted or absorbed.