The mediocrity principle simply states that you aren’t special. The universe does not revolve around you, this planet isn’t privileged in any unique way, your country is not the perfect product of divine destiny, your existence isn’t the product of directed, intentional fate. Most of what happens in the world is just a consequence of natural, universal laws — laws that apply everywhere and to everything, with no special exemptions or amplifications for your benefit — given variety by the input of chance. Everything that you as a human being consider cosmically important is an accident…
Don’t feel bad about that, though, it’s not just you. The stars themselves form as a result of the properties of atoms, the specific features of each star set by the chance distribution of ripples of condensation through clouds of dust and gas. Our sun wasn’t required to be where it is, with the luminosity it has — it just happens to be there, and our existence follows from this opportunity. Our species itself is partly shaped by the force of our environment through selection, and partly by fluctuations of chance. If humans had gone extinct 100,000 years ago, the world would go on turning, life would go on thriving, and some other species would be prospering in our place — and most likely not by following the same intelligence-driven technological path we did.
And if you understand the mediocrity principle, that’s OK.
The reason this is so essential to science is that it’s the beginning of understanding how we came to be here and how everything works. We look for general principles that apply to the universe as a whole first, and those explain much of the story; and then we look for the quirks and exceptions that led to the details. It’s a strategy that succeeds and is useful in gaining a deeper knowledge. Starting with a presumption that a subject of interest represents a violation of the properties of the universe, that it was poofed uniquely into existence with a specific purpose, and that the conditions of its existence can no longer apply, means that you have leapt to an unfounded and unusual explanation with no legitimate reason. What the mediocrity principle tells us is that our state is not the product of intent, that the universe lacks both malice and benevolence, but that everything does follow rules — and that grasping those rules should be the goal of science.