Why do people call a Psychiatrist a shrink?

All the early evidence suggests that the person who invented the psychiatrist sense of "shrink: worked in the movies. The term came about because people regarded the process of psychiatry as being like head-shrinking because it reduced the size of the swollen egos so common in show-business.

The earliest example we have is from an article in Time in November 1950 to which an editor has helpfully added a footnote to say that head-shrinker was Hollywood jargon for a psychiatrist. The term afterwards became moderately popular, in part because it was used in the film Rebel Without a Cause in 1955. Robert Heinlein felt his readers needed it to be explained when he introduced it into Time For The Stars in 1956: “‘Dr. Devereaux is the boss head-shrinker.’ I looked puzzled and Uncle Steve went on, ‘You don’t savvy? Psychiatrist.’” By the time it turns up in West Side Story on Broadway in 1957 it was becoming established.

Shrink, the abbreviation, became popular in the USA in the 1970s, though it had first appeared in one of Thomas Pynchon’s books, The Crying of Lot 49, in 1965 and there is anecdotal evidence that it was around earlier, which is only to be expected of a slang term that would have been mainly transmitted through the spoken word in its earliest days.

The name is derived from an older term, "head shrinker." "Shrink" is just a shortened form of it.

They go in hoping to have their problems shrunk.

the make your major problems seem really small!