What makes parrots talk?

Most parrots will only mimic sounds, w/o any particular understanding of what they mean.

But parrots are quite intelligent, and (some) can - with considerable effort - be taught to interact in a rational manner.

Recent study conducted from the 1980s and onwards indicates that parrots can indeed speak and understand some human speech, in rare cases on approximately the level of a 3 year old human. Irene Pepperberg's work with Alex (an African Grey) and later with Griffin show they were able to form complex responses, and relate Words to objects, numbers and verbs. Check the references below for more detail on this rather remarkable study.