Really, the exact same thing happens in a standard white painted room, the two differences being that the mirrors reflect more of the light (so that a dimmer light source will suffice to reach the same level of illumination), and that the reflection is directed instead of diffuse (this only changes the shape of the reflected light, not its amount). Maybe you can explain what's confusing you.
Think of it like dropping a ball an a hard vs soft surface. In both cases things bounce. Even on a hard floor the ball stops bouncing after a while, and in both cases the ball / floor / air get's get's warmer from the balls energy.
Seen purely as electrons -- one electron had a lot of energy, now a lot of electrons have a little. Seen purely as photos, one photon had a lot of energy, now there are lots on very low energy photons.
Overall, the collective term for this is entropy -- over time we get fewer opportunities for big photons to get created, until it's all small changes in energy and small photons -- total entropy -- and everything is background radiation.
For example, in this you have a perfect mirror, which is actually not physically possible and would mean violating several laws of physics such as thermodynamics and electromagnetism.
Another common problem is hypothesizing perfectly rigid materials or perfectly flat surfaces, which can't exist in any matter made out of atoms but which could easily beused to violate the laws of relativity.
What I'm curious about it, is there a general principle that stops things from possessing a property perfectly? For example, IIRC friction dictates that many energy transformations never convert energy perfectly, leading to far-from-perfect engines and unavoidable power dissipation in electricity transmission. Is there a similar principle that stops collisions/materials from being perfectly elastic, surfaces from being perfectly reflective, etc.? Does it go against entropy never decreasing in a system?
edit: hmm, so are the laws that dictate that perfect objects cannot exist somehow more deeply connected by a general principle (just like Noether's theorem underlies laws in various domains)?
For example, there's a recurring thought experiment about the limit of the speed of light that goes something like this: say you have a rigid rod that is one light-year long and you push on one end, won't the other end instantly move, thus proving that you can exceed the speed of light? The problem with that is that it's based on an approximate and intuitive understanding rather than a proper understanding of the physics involved. For a short rod if you push on one end the other end seemingly moves instantaneously, giving the illusion of rigidity, but in actuality what is happening is that you are transmitting forces through the rod at the speed of sound in the material, and if you make movements that are slow compared to the time and space involved then everything will appear instantaneous (since the speed of sound in steel is about six thousand meters per second). However, once you scale things up the intuitive approximation is no longer valid. What happens when you push a long rod from one end is that a displacement wave moves along the rod at the speed of sound until it reaches the opposite end, taking far, far longer to move than a signal travelling at the speed of light. Also, no material can be perfectly flat because at the scale of atoms there are... atoms, which are not flat.
That's generally the biggest reason why we can't have perfect anything, because stuff is made of atoms and atoms are messy. Often times people who let dreams of perfect materials lead them astray fail to take into account the underlying mechanism for the property they are considering (e.g. rigidity is due to forces being transmitted from atom to atom within a material). A good rule of thumb for whether or not an assumption of perfection is going to ruin a thought experiment is whether or not you're ignoring the underlying mechanism for that process. Another is whether or not you're assuming some arbitrarily small amount that you are omitting from the model because it introduces a "tax" that is annoying to account for but that can be easily bounded or instead you are assuming absolute 100% perfection that your whole model is completely reliant upon and even the slightest deviation from perfection would ruin the model.
Interestingly enough, there are a few examples of physically perfect things in the real Universe. For example, superconductors experience 0 electrical resistance to current, and superfluid helium does not experience friction internally, and electrons appear to be perfect point-like charges.
