How do people know what colour dinosaurs were?

People, in fact, do not know the colors of nearly all prehistoric animals, including dinosaurs. Recently, however, scientists have found a technique to determine the colors of a fossilized feathers by examining them with a microscope. Therefor, the colors of a two, early bird-like dinosaurs, are as follows:

Confuciusornis (an early bird, similar to Archaeopteryx): Feathers show regions of gray, red/brown, and black.

Anchiornis: For this, you would probably like to see a picture, showing it just as it would have looked in real life (except the tail, whose colors are still unknown). You can see a picture of an Anchisornis in its true colors by going to the related link.