There is an Irish legend (with some variations...) speaking of Leprechaun, an elf you will find ath the end of the rainbow, whom you may force to disclose where does he keep his cauldron, full of gold coins... Simplified version states that you will just find the cauldron at the end of the rainbow.
NIce, isn't it? Well, when you check a couple of physics' topics (specifically, optics), you may find that it is even more beautiful...
Isaac Newton proved that white light is the result of the overlaying of "all" colours, that could also be separated by means of a glass prism. This effect is caused by the different behaviour of each wave length when moving from a carrier (air) to the other (glass), or vice-versa.
You may repeat this effect at home, with a glass prism. Or any other glass object, with polished, flat surfaces. When lit with white light, you will see how light is split into basic colours.
But, what happens if the object is a sphere? - well, basically the same, but the (deviation) effect might be stronger for specific colours or wave lengths, depending on the angles.
And what if the sphere is made of water? Then, the effect is similar; light rays change direction at the contact surface between air and water. Angle might vary from the reference glass one.
Now, let's go to the "rough" part. Falling rain takes spheric shapes. Forget about the romantic droplet shape you always drawed. Sphere is the most efficient shape for fluids in free-fall...
From that point, imagine a situation where rain is happening somewhere between you and the sun. There you should be able to see the rainbow.
If droplets were very small (which is the case, due to the distance), what you see is just the change in the angle and colour separation. But each droplet deviates light in given directions, so you will only see "one" colour out of each droplet...
Of course, you will see the other colours from surrounding droplets. Relative angle to the sun will be different - and this is the key point.
Up to here, I just described a physical effect on the angle between the sun, the droplet... and your eye. This angle can appear in a vertical plane, or horizontal, or in any other inclination.
Thus you may find a "circle of droplets" that matches a specific angle value, and thus will "send" the same colour to your eye.
If you change this angle slightly, your eye will see a different colour. The new droplets affected will be concentric to the previous ones.
Circle by circle, your eye will receive information from all droplets in the water "curtain" placed between you and the sun. You see all different colour circles - you see the rainbow.
Of course, borders are blurred for each colour. This is caused by the whole water mass falling down, not just as an ideal "curtain".
There is more to say! Read on...