Simply stated, it means that most people aren't special (especially those who think that they are). It can also be extended to things and events.
It is named after Copernicus, because he is famous for pointing out that the earth is not the center of the universe. By definition, most observers are "typical" (look up bell curve). Hence, most people are just average Joes and Janes. Another way to look at it is to realize that there is no center to the universe, and even if there were, you probably aren't it.
So when someone complains about the behavior of others, or how society would be better off without a certain demographic, or other such self-centered talk, it is worth pointing out that not only are they not alone in this world, they aren't even close to being anything special in it. Graveyards are full of indispensable people.