All you have to do is take some transparent paint (watercolor will work) and start coloring. If you are asking how would people know what colors to use, they can use whatever color they want. Obviously the sky is blue and the grass is green, but you can make educated guesses on other things like eye color or clothing.

I'm not sure /r/AskHistorians is the best place to ask this question, unless you're wondering about how coloring a photo was done in the past. /r/Colorization is probably a good place to check out. They have some tutorials in the sidebar.