I know it originates from "head shrinking", but it doesn't help me a lot to understand the etymology. Why are psychiatrists called that? Is it like "my head is swollen [from anguish, misery, stress, etc.], I must have it shrunk"?
The word indeed comes from "head shrinker", and likely originates from the "shrunken heads" of tribal rituals. One possibility from World Wide Words:
All the early evidence suggests that the person who invented the psychiatrist sense worked in the movies (no jokes please). We have to assume that 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. Or perhaps they were suspicious about what psychiatrists actually did to their heads and how they did it and so made a joke to relieve the tension.
I'm afraid I have to disagree here. From my understanding, and a recent article in the Atlantic, derived from the new text Marketplace of the Marvelous: The Strange Origins of Modern Medicine, referring to a psychiatrist as a shrink refers not specifically to head-shrinking tribesmen, but to the field of phrenology, a significantly closer cultural institution to psychiatry.
"The national obsession with head size and shape also infected daily conversation. Many modern phrases trace their roots to phrenology, including “highbrow” and “lowbrow,” “well rounded,” and “shrink” (as in “shrinking” certain undesirable qualities). “Getting your head examined” also has phrenological roots. Though generally considered an insult today, in the past, it was just what most people wanted. By the mid-19th century, the Fowlers’ publications could be found all over the country, and phrenological ideas had become a part of everyday conversation."
To answer your question, reading up on phrenology in general can explain a number of the specific phrases we use for psychiatry, since phrenology dealt directly with size of different parts of the head as means of attempting to quantify different attributes.