What process makes the sun's heat light?

I assume you mean convert the sun's heat energy into light, this process occurs in almost every material in the universe, including yourself, the sun's 'heat' is merely particles bouncing off one another very fast like thousands of bouncing balls in a huge container that is shaken up, when these particles collide they sometimes give off a bit of energy, we call this energy electromagnetic radiation.

You can't see most radiation as it is beyond the visible spectrum, infra-red is the most common radiation emmited by hot objects, this is the radiation used to see in the dark by night vision goggles. When objects get very hot like the surface of the sun, they begin to emit higher energy radiation, like visible light, if you've ever seen a glowing hot metal you've seen first hand how a high temperature object can emit visible radiation.

The process howver is very complicated and is still not fully understood but to simply it each particle has an outer shell made of electrons, when these shells of electrons collide they change energy states and release bits of radiation as they change energy states, for a full explanation i would recommend you read about quamtum physics and electron energy states.