To see why rainbows are curved it helps to understand what causes them in the first place. We see rainbows when rain is falling in front of us and the sun is shining behind us.
When sunlight enters raindrops most of it passes straight through. But some of the light strikes the drop at such an angle that it's separated into the rainbow colors and reflected back out the front of the drop. Each color is reflected at a slightly different angle, but all between forty and forty-two degrees.
The narrow range of angles at which the colors are reflected is the key to why rainbows must be curved. The next time you see a rainbow notice that you are in the center of the arc, and that if you move the rainbow moves with you. This is because even though all the raindrops out there are reflecting rainbow colors, they do so in very narrow beams. You see the colors from only those drops whose beams shine on you.
The only shape that will allow the light beams to shine on you at the precise angles is an arc or a circle with you at the center so that you are always the same distance and angle from any point on the arc.