I take "special" here to mean "deserving of privileges over and above other instances". In this sense, if you think you are "special" that means you think that you ought to be able to bend the rules more than other people, or that cashiers ought to be nicer to you than to the rest of the people in line, or that you deserve to get that last discount TV on the shelf on Black Friday, or that it's okay for you to zoom past traffic on the shoulder because you're in a hurry moreso than anyone else.
Possibly part of your trouble in interpreting the essay is that you're missing out on what the essay is in response to. The question was posed: "What scientific concept would improve everybody's cognitive toolkit?"
Whether the Mediocrity Principle is "correct" or not is not even the point. The point is that assuming the Mediocrity Principle to be true and acting as if you and your situation are not special is a productive stance to take in life, as it is in science.
It's another way of saying "the world does not revolve around you", "you are not the center of the universe", etc.