Say you have a Faraday cage mesh with holes small enough to reflect the radio waves. What happens to radiation when it is emitted from inside the Faraday cage?

Does it keep reflecting off of the inside of the cage forever?

No. Unless the cage is a perfect conductor (which it is impossible in real life to make or find), a portion of the energy of the waves will get absorbed by the walls of the cage on each reflection, eventually causing the wave to attenuate or "die out". Given the incredible speed of light, these reflections happen close to a billion times each second in a cage around $$1\, \mathrm{m}^3$$ in volume, so the attenuation happens really quickly.
Typically, the radiation penetrates a characteristic distance called the "skin depth" inside the material of the conductor, before falling to around $$1/3$$ of its initial strength. For example, copper has a skin depth of around $$8.5 \, \mathrm{mm}$$ for a radiation of frequency $$60\, \mathrm{Hz}$$. The conductor will get heated up as the wave is attenuating; it is a continuous process.