To put it more simply: many birds lose and re-grow their feathers twice a year. Not quite all at once (that would leave the bird naked), but over the course of a few weeks. This is known as molting, and it often occurs just before migration. So in the spring, just in time for breeding season, birds have all new feathers. If you know anything about how evolution works, it should not be a big surprise that some birds' new breeding-season feathers are spectacularly colorful, while their winter feathers are drab and provide much better camouflage. A good example of this is the male mallard: in the spring the males are quite colorful, while in the early winter they look almost exactly like the females.
There are lots of variations and complications. There may be more (or fewer) than two molts a year; there may be more than one explanation for a bird that's not in breeding plumage during breeding season (it might be too young, it might just be molting late, etc), plumage may change for males, females, or both; the timing of molt may not match your expectations of the season (the bird might migrate between northern and southern hemispheres, or live in the tropics year-round), and so on. As a result, there's a lot of vocabulary around molting and plumage and birds' ages, which can trip up even very experienced birders. Luckily, many birds have an easily-recognized "breeding plumage" for at least one sex. A bird (of that sex) of such a species, seen in any other plumage, is safely described as "non-breeding."