Why do rainbows always appear to be far away from us?

When sunlight enters a (roughly) spherical drop of water, it will be refracted at the entrance point, perhaps reflect one or more times off the interior surface of the drop, and then will be refracted again at the exit point. A significant proportion of the light enters, reflects once internally, and then exits, leaving at an angle of about 41° from the incoming ray. For a single drop, the light that follows such a path will emerge in a cone, centered on the sun and with a half angle of 41°.

When the light refracts at the air/water interface, the angle is wavelength-dependent. The result is that the cone is actually a conical rainbow, ranging from an inside blue cone at an angle of 40° to an outside red cone at an angle of 42°.

Of course, you can't see the rainbow from a single water drop, because you only see light that makes it to your eyes. So, the color (if any) you see from any one drop is the part of that drop's rainbow cone whose light hits your eye. The result is that your eye sees colors coming from a reversed cone of drops; drops at an angle of 42° between you and the sun shine red, those at an angle of 40° shine blue, and those between shine intermediate colors.

Now, why can't you see the cones of color as actual cones? Because the cones' apexes are all at your eye, so there's no way for your eye to see the depth of the cone.

And why does the light always bounce exactly once internally? Answer: it doesn't. Some light bounces twice and comes out at a slightly different angle, forming a double rainbow. Other numbers of bounces form other rainbows, but they're generally difficult to see.