The following is from Richard Dawkins' book Unweaving the Rainbow.
Raindrops have a more complicated effect than Newton's prism. Being roughly spherical, their back surface acts as a concave mirror. So they reflect the sunlight after refracting it, which is why we see the rainbow in the part of the sky opposite the sun, rather than when looking towards the sun through rain. Imagine that you are standing with your back to the sun, looking towards a shower of rain, preferably with a leaden background. We shan't see a rainbow if the sun is higher in the sky than 42 degrees above the horizon. The lower the sun, the higher the rainbow. As the sun rises in the morning, the rainbow, if one is visible, sets. As the sun sets in the evening, the rainbow rises. So let's assume that it is early morning or late afternoon. Think about a particular raindrop as a sphere. The sun in behind and slightly above you, and light from it enters the raindrop. At the boundary of air with water it is refracted and the different wavelengths that make up the sun's light are bent through different angles, as in Newton's prism. The fanned-out colours go through the interior of the raindrop until they hit its concave far wall, where there they are reflected, back and down. They leave the raindrop again and some of them end up at your eye. As they pass from water back into air they are refracted for a second time, the different colours again being bent through different angles.
So, a complete spectrum -- red, orange, yellow, green, blue, violet -- leaves our single raindrop, and a similar one leaves the other raindrops in the vicinity. But from any one raindrop, only a small part of the spectrum hits your eye. If your eye gets a beam of green light from one particular raindrop, the blue light from that raindrop goes above your eye, and the red light from that particular raindrop goes below. So, why do you see a complete rainbow? Because there are lots of different raindrops. A band of thousands of raindrops is giving you green light (and simultaneously giving blue light to anybody who might be suitably placed above you, and simultaneously giving red light to somebody else below you). Another band of thousands of raindrops is giving you red light (and giving somebody else blue light...), another band of thousands of raindrops is giving you blue light, and so on. The raindrops delivering red light to you are all at a fixed distance from you -- which is why the red band is curved (you are the centre of the circle). The raindrops delivering green light to you are also at a fixed distance from you, but it is a shorter one. So the circle on which they sit has a smaller radius and the green curve sits inside the red curve. Then the blue curve sits inside that, and the whole rainbow is built up as a series of circles with you at the centre. Other observers will see different rainbows centred on themselves.