Why does a precipitate form in a supersaturated solution?

A precipitate forms because the solvent has dissolved the maximum amount of solute and can no longer dissolve the excess. The excess therefore forms a precipitate. You can think of it this way. Imagine you are cleaning up juice that you spillt with a sponge. After a while the sponge becomes so filled with juice that it cannot soak up any more, or in other words the sponge becomes saturated with juice. In a similar way, after a while a solvent will no longer be able to "soak up" any more solute and the solute will remain as a precipitate.