The sun heats the earth through radiation. Since there is no medium (like the gas in our atmosphere) in space, radiation is the primary way that heat travels in space. When the heat reaches the earth it warms the molecules of the atmosphere, and they warm other molecules and so on. This heat transfer is called conduction. It also keeps the Earth warm at night when the sun rays are not available. This is in contrast to space where heat can only exist where the heat waves are traveling, leaving huge temperature differences between a place in the sun and a place in the shadow.

The sun emits light, and light is energy. The light travels to the Earth, and is absorbed into it, generating heat. It works in much the same way as a fire does in a fireplace - the light (most of which you can't see, because it's too long wavelength) heats up the room>