First of all, any time you ask a question like "what is really happening in situation X", the correct answer is always "no one can really say." But what you probably mean is "how does theory Y provide insight into what is happening in situation X", then we can give a good answer (probably a few!). Since you used the phrase "at a quantum level", I surmise that you want an answer involving the theory of quantum mechanics, but even then it's not so obvious what "theory Y" is here-- because we have ordinary quantum mechanics, and relativistic quantum mechanics, and quantum field theory, etc., and they all give different answers to your question! But I'll give an answer from the most ordinary quantum mechanical point of view I can.
Quantum mechanics basically takes the wave mechanics of classical physics and applies it to individual particles. There are a few differences that crop up when you do this, but I think the main concept here carries over pretty well-- the concept of interference. In classical wave mechanics, we have Huygens' principle, which says that every part of a wave acts like sources for how the wave evolves forward in time. Also, we have the "superposition principle", which says that every solution to the wave equation that comes from one source just adds up with the solutions that come from all other sources. This means in classical waves, the waves get a high total amplitude wherever there is constructive interference, and low total amplitude wherever there is destructive interference.
In quantum mechanics, we also have a superposition principle, except now it applies to individual particles. It basically says that anything that we can describe as something that can happen to an individual particle can happen in superposition, so what 'actually happens" is a kind of constructive sum over all these "possible happenings." This is the spirit of the Feynman path integral approach, for example. So using this mathematics, what has a high probability of happening is what receives constructive interference over this sum, and what has low probability is what destructively interferes. This is a key point-- each individual term in the sum starts out equally likely, even ones that correspond to absurd behavior, but the absurd behaviors cancel each other out, sort of like monkeys voting in an election.
In the case of reflection, we find that the presence of the mirror allows a certain behavior to receive constructive interference, which would not if the mirror were not there. The new behavior is "angle of incidence equals angle of reflection", and the mathematical property of that type of solution is that it exhibits "stationary phase." Stationary phase means the phase of the type of process (so how quickly the amplitude of the process varies over the set of very similar processes) is not varying over the physically allowed process, but does vary rapidly as soon as you test a non-physical process. Thus, the "angle of incidence equals angle of reflection" gives an extremum in the phase as you vary over a range of possible processes-- in this case, it is the minimum time to get between specified points A and B. Without the mirror, the only minimum time is the straight line between them. With the mirror, a second possibility emerges, the local minimum in time that comes from an equal angle of incidence and angle of reflection path between A and B that glances off the mirror. (It takes more time than the straight shot, but less time than all its neighboring paths, so it exhibits stationary phase and so constructive interference when you add the neighboring amplitudes.)
This still doesn't answer what the mirror is doing that allows for this new stationary phase solution. The classical answer to that is that the mirror acts like a source of waves that cancel the incident wave within the mirror, and constructively interfere to make a reflected wave. Quantum mechanically, the mirror creates a boundary condition on the photon wavefunction that forces the wave function to go to zero at the surface of the mirror, and this constraint suffices to give the reflected wave when you apply the superposition of wave functions analyzed in terms of all the modes that obey that constraint and have the energy of the incident wave. You could even do it with quantum field theory, and one way to picture that would be to say that the incident photon is destroyed by the mirror, but its energy must be accounted for, and since the mirror is elastic, it must use the energy to promote a "virtual photon" to the status of a real photon. What's more, the virtual photon not only has to have the energy of the original photon, it must also have a wave function that resonates constructively with the original photon-- in other words, it is indistinguishable from the original photon, so is ruled by the same wave function, and that wave function must experience constructive interferece (for all the above reasons) to have a reasonable probability of actually happening.
So you may be surprised to find there is not just one answer to your question, and there might be new possible answers in a few more centuries, but the key idea is constructive interference set up by the way the mirror is required to prevent the photon from crossing its flat surface, yet energy is also required to be conserved. Generally, we're happy if we have one way to think about it that gives good results and seems simple enough for us to understand.
