Why do birds have 3 types of feathers?

It depends on the bird. Quite often the most colorful displays are because of sexual selection and evolution. When mates go for birds with best and most pretty feathers, they tend to have offspring with very pretty display feathers. And the cycle continues with mates selecting for mates with good feather displays, and good feather displays allowing mating and offspring with good feather displays.

With sexual selection, once the cycle starts, evolution takes over and the whatever criteria that is being selected for is hyper accentuated by evolution, because the organism with that does the best because they get to mate, and their kids also are quite likely to share that same quality.