What causes rainbows to appear after rain?

Rainbows form when sunlight enters water droplets still suspended in the air after rain. As light enters a droplet, it slows down and bends (refracts), reflects off the inside back surface of the droplet, then refracts again as it exits. This process separates white sunlight into its component wavelengths because each color bends at a slightly different angle, with red bending the least and violet the most. You see a rainbow as an arc because you're observing millions of droplets simultaneously, each sending a specific color to your eye at a specific angle, roughly 40 to 42 degrees from the antisolar point directly opposite the sun. This is why you need the sun behind you and rain in front to see one.