Basically, LEDs are, at their core, two different parts of a junction (hence a di-ode) where one side can only take electrons with a certain amount of energy. The electrons coming into the other side of the diode have too much energy, so the excess energy is released as light. (the common metaphor is a hole and the electron is too large to fit, so it has to shrink size).
Interestingly, this is the same principle as photo cells, only in reverse. If you run power into a solar panel, under the right circumstances, they will emit light.
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