The first thing to think about is how rainbows are formed. Imagine a big wall of water, made up of water droplets, in front of you and off in the distance. Each water drop is a tiny mirror which reflects sunlight (that comes from behind you) and also separates the sunlight into different colors. In order for you to see the rainbow, light from the sun must reflect at just the right angle to reach your eye. You see the different colors in stripes because red light reflects a little bit differently than blue light.

Obviously, not all the light from the sun reflects right into your eye. What happens if you move 100 feet away from where you are? Can you still see the rainbow? If you do move to a new location and look at the rainbow again you might find that it has changed, maybe getting brighter or more faint. This is because you are looking at a different part of the wall of water drops and maybe the wall is a better or worse mirror in different places.

Now, finally we get to your question: Why is a rainbow bent. Well, remember that the angle between you, the water droplet, and the sun has to be just right to see anything. I believe that you only get that correct angle on a ring on the sky. You only see half of the ring because the other half is below the horizon. All the other sunlight that is reflected by that wall of water goes elsewhere, to be seen by someone else.

Thanks for the great question and let me complement you on your great intuition.

You are correct in guessing that the reason rainbows are bent is due to moisture in the air. I'll try to give you a more detailed discussion of this amazing feature of rainbows.

First, let's consider just a single drop of water and how light travels through it. As light enters the raindrop, it is refracted (the path of the light is bent to a different angle), and some of the light is reflected by the internal, curved, mirror-like surface of the raindrop, and finally is refracted back out the raindrop toward the observer.

I've tried to attach a picture to help show the following discussion. This schematic represents the path of one light ray entering a raindrop at point A. As the light beam enters the surface of the rain drop, it is bent (refracted) a little and instead of continuing to point D, strikes the inside wall of the raindrop at point B, where it is reflected back to point C. As it emerges from the raindrop, it is refracted (bent) again into the direction E. The angle created at point D is 42 degrees.

The ray drawn here is significant because it represents the ray that has the smallest angle of deviation of all the rays that can enter the raindrop. This means that much of the sunlight that is refracted and reflected through the raindrop is focused along this path. This is known as the rainbow ray.

Since the various colors that make up white light all have slightly different wavelengths, each color becomes slightly separated from the others as the light ray is refracted and reflected. So, rays that strike the raindrop at this angle of 42 degrees will tend to form a concentrated, strong beam in which the colors will be widely separated. This creates the color bands in the rainbow, with blue along the inner portion of the bow, and red on the outside edge of the bow.

Since the raindrop is circular, the reflection it creates is also circular. We don't see a full circular rainbow however, because the earth gets in the way. The lower the sun is to the horizon, the more of the circle we see. At sunset, we would see a full semi-circle of the rainbow with the top of the arch 42 degrees above the horizon. The higher the sun is in the sky, the smaller the arch of the rainbow above the horizon. If the sun is more than 42 degrees above the horizon, no rainbow will be visible.