What process predicts that black holes can emit thermal radiation?

The process predicting that black holes can emit thermal radiation is called Hawking radiation.

Stephen Hawking proposed the effect in 1974 by applying quantum field theory to curved spacetime near an event horizon. In this description, a black hole is not perfectly black: quantum effects allow it to emit radiation with a temperature determined by its mass.

The temperature is inversely proportional to mass. Large astrophysical black holes are extraordinarily cold and emit far less radiation than the cosmic microwave background. A tiny black hole would be hotter, lose mass faster, and eventually evaporate in a powerful final stage under the theory.

Hawking radiation is not ordinary light escaping from inside the horizon. It arises from quantum fields in the curved spacetime outside and near the horizon. The prediction also creates the famous black-hole information paradox, because evaporation appears to challenge the principle that quantum information is preserved.