A rainbow doesn't exist in a single position. If you and I stand side by side and look at a rainbow, we'll aactually see it in a slightly different place. This is why they seem to move away from you as you move towards them. The light reflected in raindrops is split into the different colours and also re-directed. This means that the angle of any rainbow to your point of view is consistent. In fact every part of it is at the same angle from a point centred at the back of your head and the line between that point and the shadow of your head. Hence, it's a circle. Because it can't appear below the horizon, you see an arc. If you're in the mountains and conditions are right, you may see a full circle (a Brocken Spectre).

I probably can't explain it better than it is explained on Wikipedia so I suggest you look there.