Why do photons excite electrons?

Sorry, I wasn't being very clear. What I mean is, why do photons interact with electrons?

What we have discovered up to now with our studies in physics is that there exist 4 fundamental interactions of elementary particles.

Both the photon and the electron are elementary particles and interact with the electromagnetic interaction.

Now the electrons can be free , as for example in an accelerator beam, or bound with the electromagnetic interaction in an atom, as in the hydrogen atom.

If they are free, a photon hitting them will scatter elastically, or might give up part of its energy to the electron and go away with a smaller energy/frequency ($E=h\nu$).

If bound, it is in an energy level about the nucleus which has a unique, quantized energy, $E_1$. Over it will be unoccupied energy levels . An incoming photon, if it has an energy that corresponds to the difference between an empty energy level $E_2$, i.e. it has energy $E_2-E_1$ can transfer its energy to the electron kicking it up and disappearing as an individual photon . The electron will probably decay from that energy level emitting a photon of energy $E_2-E_1$ but it will be a different photon. In nuclei with large $Z$ there may be cascades of photons if the energy of the initializing photon is large and there exist intermediate energy levels.

Now if we go to second quantization, the photon interacts with the electron because it is the carrier of the electromagnetic force. This covers both the bound and the unbound state of the electron, except that in the bound case still the energy has to be $E_2-E_1$ to give a large enough probability of interaction.

Photons are electromagnetic waves that propagate in wave packet. Those wave packet carry a defined quantized amount of energy.

When a photon interact with an electron it will give away its energy to the electron. The electron will have more energy and hence a larger velocity. This mean that the electron will indeed "orbital" further away from the nucleus (the probability density of finding the electron will be higher further away from the nucleus).

If the new orbital of the electron is an allowed one, the electron might stay on this orbital for a while and then decay back to its ground state position. At that point the electron will emit a photon to give away the extra energy.

If the new orbital of the electron is not an allowed one (Quantum mechanic tells you which orbital is allowed) we then say that the electron goes into a virtual states (a state that does not exist) and immediately re-emit the photon.

You can find a mathematically more involved description here: http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/photel.html