Why are the Northern Lights colorful?

Because that's how physics works.

They're colorful for, to a first approximation, the same reason a neon sign is colorful: the energy being emitted is limited to a specific value, and the wavelength corresponding to that energy happens to lie in the human visual band. Since it's only one wavelength rather than a broad spectrum, we perceive it as a specific color.

The reason there are different colors is that different transitions are taking place, and each of them has a particular energy (and therefore wavelength of light) associated with it.