Some of that energy turns into heat every time the photons hit something, but heat is a mix of kinetic energy and light so you do get a fraction of that light bouncing around indefinitely, it's just not in the visible spectrum.
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.
Yeah... Warmer is kind of mixing metaphors since warmth is just a statistical aggregation of stuff moving around. The point is that photons result from electrons changing energy levels and disappear when they hit an electron and change its energy level. A big change makes a high energy photon which might bump into an electron, resulting in a higher energy electron and a new, less energetic photon. This keeps on happening.
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.
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.