What causes a rainbow to appear after rain?

A rainbow forms when sunlight enters water droplets suspended in the air and undergoes refraction, internal reflection, and refraction again as it exits. When white sunlight enters a droplet, it slows down and bends, separating into its component wavelengths because different colors refract at slightly different angles. Red light bends the least and violet the most. The light then reflects off the inside back surface of the droplet and exits, spreading the colors into a spectrum we see as a rainbow arc. You see a rainbow after rain because the air is still full of water droplets while the sun is shining, typically when the sun is behind you and the rain is in front of you at the right angle.