The title really says it all. How can a fundamental partical which cannot be broken down any further release another fundamental particle. If this is a stupid question let me know and I shall do more research.
Depending on the level you need, I can think of it in two ways:
Undergraduate: Imagine a boat and a fisherman in the middle of the lake with very steady water - no waves on the surface. The fisherman catches a fish, the rope starts wiggling and creating waves on the surface of the pond. Does it mean that the rope consists of waves? No, the rope creates perturbations in medium (water), which we observe, but it tells nothing about internal structure of the rope, it rather tells about its dynamical state. In the same way wiggling electron (antenna) perturbs space around itself and creates em-waves, hence, photons, which characterize only electrodynamical state of the electron, but not its internal structure.
Graduate: Well, you might want to have a specific QED process in mind. A free electron propagating with constant velocity in empty space cannot release photons, it violates energy-momentum conservation. For example, in electron-positron pair production, electron releases a photon in vicinity to a heavy nucleus, which is related to the structure of QED vacuum (medium, "water of the lake"), not an electron.
An electron, left to itself, does not emit a photon. You can get it to emit a photon only by acceleration (application of force), or by interaction with other matter.
The emission of a photon has to conserve energy, spin, momentum, charge... but it needn't destroy an electron to do any of that. Turning the electron spin upside down, though, is a +1 or -1 spin change, which is just right for conserving angular momentum when emitting a spin 1 photon. This means that one particular electron may be a key to a process emitting a photon.
A photon is energy (in the form of electromagnetic radiation). An electron releases energy when it's energy level changes by the amount equal to the energy of a photon. When an electron drops down an energy level, for example, it can release a photon of that energy. That's fluorescence, for example. The energy released, and therefore the energy (and wavelength) of the photon, depends (in this case) on the energy levels, and electrons can release photons up into the X-ray band.
Photons are not particles in the same way as electrons. They are very commonly created and destroyed (turned into heat, for example).
If an electron is a fundamental particle how can it release photons.
An electron is a fundamental particle, it has no interior structure at all, as far as we know. It does have an electric charge, but the vital point is that the photon itself does not have an electric charge, so when an electron emits a proton, it is not giving away electric charge, it still remains fundamental.
When it emits a photon, it is giving away momentum, say to another electron near it, but losing momentum does not effect it's basic "fundamental" nature in any way.
There are no parts coming off of it, nor is its intrinsic spin affected by the process of emission. An electron is not made of photons, and because they do not carry away electric charge, it can absorb and re-emit photons from other electrons all day long, which it normally does, btw.
If an electron is not free, but instead is part of an atom, it can drop down to a lower level, nearer the atomic nucleus, by emitting a photon, until it reaches the ground state, the lowest level it can be at, then at that stage it will not emit any more photons.
I do apologise if you know all this already, but being fundamental does not mean it loses anything of itself by emitting a photon, it just gives up some energy of movement to another particle.
Another way to the ones already stated:
Elementary particles in the elementary particle table of the standard model of physics are of two kinds , fermions and bosons.
General rule: Fermion number is conserved, whereas boson number is not conserved. Given the energy bosons can be created without problem, as long as quantum numbers are conserved. That is why the number of photons is not a conserved quantity.
The word "elementary" refers to the role these particles play in the mathematical representation of the model. They are necessary building blocks. There exist further qualifications, like spin and other quantum numbers, due to their role in the model.
If an electron is a fundamental particle how can it release photons
Because it isn't really fundamental. We can make electrons (and positrons) out of gamma photons in pair production. Then we can annihilate the electron and the positron in annihilation. We can do the same sort of thing for other particles too:
Image credit CSIRO, see The Big Bang & the Standard Model of the Universe
If this is a stupid question let me know and I shall do more research.
It isn't a stupid question. It's an intelligent question. Another intelligent question is this: Where do the fundamental quarks and gluons go in low energy proton-antiproton annihilation to gamma photons?