Electrons flowing through a LED emit light as they travel through a certain part of the diode. It involves quantum mechanics, but the electron jumps from one energy level to another and the lost energy produces a photon. A photon is a "particle" of light.
Incandescent bulbs emit light because the filament is hot. This actually emits light from electrons changing energy levels as well, but it's heat that puts the electrons in the higher energy states so they can drop back and emit light.
Gas discharge tubes work like LEDs in some sense, but they have to heat the gas up to high temperatures before the current will flow, so they have to be hot to start with.
All of these forms of light emitter heat up due to current flow, light that doesn't make it out of the device, and other forms of energy loss.