The rainbow is fairly easy to understand in terms of particles. Many people have explained this phenomenon excellently before me, so I’m not going to pretend originality on this point.
This is merely another rehash of an old and well established interpretation.
The first thing to note about the rainbow is that it is a reflection of light coming in from behind the observer.
Furthermore, in order to diffract into multiple colors, the light has to enter the medium that it is reflected from. The reflection is off of the inside of the medium, not its exterior surface.
In the case of the rainbow, the medium is water in the form of spherical droplets.
Only light that penetrates into the droplet before reflection can produce the rainbow effect. Light that reflects off of the exterior of a droplet will not diffract into multiple colors. Light that goes straight through the droplet is lost to the observer.
The type of reflection that is required in order to produce diffraction with the use of water droplets is very peculiar and rare. It only happens at two precise angles relative to the observer.
We have the rainbow and the double rainbow, but not a third or a fourth rainbow.
To illustrate why the reflections required are as rare as they are, we must consider the raindrop, and the refraction and reflections that are required for the light to make the path from behind the observer, through the raindrop, and then onto to the observer.
Observer, rainbow and light reflecting off the inside of a raindrop
For light to enter a raindrop, it has to hit it at a near perpendicular angle. However, for light to reflect, the angle has to be slant.
Light has to enter the raindrop, and subsequently make multiple slant reflections before leaving. Any other scenario will either see the light fail to reflect towards the observer, or fail to diffract.
This, in short, is how the rainbow works and why we only have two and not a multiple of them.