Joseph Kirby asked:
How can evolution explain that birds with such amazing colorful feathers were developed?
Answer by Craig Skinner
Traits evolve in a breeding population if they are adaptive. A heritable trait, randomly arising in an individual, will spread if it improves the chances of the individual surviving or reproducing. In short, natural selection of favorable variations.
But how can bright plumage be favorable? At first sight it seems maladaptive, making the bird easier for predators to spot.
The answer, in general, is that female birds find bright plumage attractive so that bright males have a reproductive advantage and leave more offspring.
This is an example of sexual selection (described in detail by Darwin) where traits arise in one sex (usually males) which are either attractive to females (as with bright bird plumage) or help males to compete for females (as with antlers in deer).
Sometimes, there are other explanations for bright plumage. Field work has shown that some birds benefit, not because of female preference, but because the plumage signals to competitors that a territory is occupied. And in yet other cases, the chemistry that makes the plumage bright makes the flesh taste bad, and predators learn to avoid these birds and concentrate on drabber birds.
Finally, bird eyes have four types of cone as compared with three in humans. Hence birds have much fuller colour vision than us. In particular they can see in the ultraviolet part of the spectrum which is invisible to us, so that birds whose plumage looks dull grey or brown to us may be seen by other birds in gorgeous colours that we cant even imagine.
This is an active research field and there is much more to be said about bird plumage both in males and in females. But explanation is on standard, modern evolutionary lines (natural and sexual selection, genetics and epigenetics).