Think about how white light going into a prism makes a rainbow. This is because the different wavelengths of light are bent different amounts and so emerge from the prism at a slightly different angle. Raindrops work sort of like that prism, splitting light up into various colors emerging at different angles.
For a particular droplet of water depending on where you are standing you would only see light of a particular color coming from it. It is splitting light into the full spectrum but when you view it from a specific angle you are only seeing the one color. Other droplets elsewhere in the sky viewed at a different angle will show you a different color.
Because this all depends on the angle at which you view those droplets of water, a particular color is going to be visible at a specific angle from the sun. This applies to any angle from the sun which means a rainbow appears as a circle opposite the sun at about 42 degrees. It only appears as an arch because some of that light is being blocked by the ground; if you were flying in an airplane you can see the whole circle.
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