I have always been told and understood that rainbows occur from water droplets and the light gets dispersed into its colors. I also understand that light travels in straight lines. This might sound like a stupid question but it most likely has a simple answer: Even though light travels in straight lines, why do rainbows appear to be bending?
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You're right, the rainbow is the dispersion of white (multicolored) light into its components because each wavelength refracts on a different angle. (Or in other words: The angle of the refraction depends on the wave's wavelenght).
Now what is also important is that in order to see a rainbow, you'll have to look at the exactly right angle at it. Of course there is not only one angle where the rainbow is visible, it is an interval. (Other people can see the rainbow too, right?) To be more clear: The incoming (sun's) rays can be approximated to come from a point source and enter the atmosphere at a certain angle. From there, they come onto the water droplets at a certain angle. On the droplets, they refract at a certain other angle. And then they have to enter your eye so you can see it.
So for you to see it, you'll need to be observing exactly there, where all these conditions are satisfied. A 'rainbow' (more like the colors of the rainbow) can be seen at every point where those conditions are satisfied. If you'd draw a straight line from your eye to the droplets and wanted to keep the angle between the line and the droplets constant (which is necessary so you can see the rainbow), you would get a circle. (A cone, if you connect all the dots: The line from the eye and all the points where the angle is constant.)
To summarize: All the droplets from where the sun's light would refract and could be seen by your eye as a rainbow will be on a circle. Unfortunately, the circle is 'too big' and cut off by the horizon; Thus, you'll only see a part of it - the bow.
Edit: This answers why there is a bow. Now let's have a look at why there isn't a line: In order to have a 'rainline', you'd need light incoming from a line source, so that the refraction from the droplets can reach your eye. This is not the case with the sun, which is a spherical source.
Hope this helps.
