I've never heard the term before, but in context I'd guess it means something like "an explanation that implies we're less important than the previous explanation did". Heliocentrism vs. geocentrism, evolution vs. a supernatural creation narrative culminating in people, etc.
Freud's psychoanalysis has been often put in the same category of "Copernican" things as heliocentrism and evolution.
As per a recent comment this thread is meant to voice contrarian opinions, that is anything this community tends not to agree with. Thus I ask you to post your contrarian views and upvote anything you do not agree with based on personal beliefs. Spam and trolling still needs to be downvoted.