Within the anthropic principle is the rejection of the idea that we require an extraordinary explanation because we are here. The universe is here anyway: it doesn't care whether we're here or not (and it has its own anthropic principle, but I'm talking about the biological, not the cosmological one). If we weren't here, we wouldn't be inquiring. Sentience leads to inquiry. Sentience being unlikely doesn't preclude natural explanation.
The Fermi paradox is that, if life can reasonably arise, why don't we have evidence of any other life? It's not technically a paradox, as it is not assumed that life must be so likely that civilizations develop close enough to each other to be able to initiate contact.
I want to apply the anthropic principle to the Fermi paradox and arrive at conclusions that are even MORE counterintuitive. For example: We're an isolated civilization, which by definition means we haven't encountered any Others. But maybe isolated civilizations are improbable! Just because this facet of our existence might be improbable doesn't mean that we've failed to recognize nearby complex life (or that our lack of contact requires extraordinary explanation). Civilizations that develop with a great deal of contact with their Others, will never worry about this problem. We might be a rare intelligence to have to wonder about this. Just because we're wondering about it doesn't mean our experience is typical.
To turn this again on its head: If we suppose that, due to the number of stars in the universe, individual intelligent life forms (e.g., specifically Homo sapiens) are improbable, but some kind of intelligent life is inevitable, and that there are many, perhaps billions of such life forms, then it may be possible or even likely, that despite most of them being isolated, that a select few may have contact with each other. THESE rare and fortunate intelligent beings may thereby initially draw false conclusions that intelligence is common. Again, they must also apply something like the anthropic principle if they wish to secure objectivity.
Aside: We suppose that life must be carbon based and oxygen breathing. But why? Maybe life in other star systems that has life analogous to our plant life, produces something other than oxygen out of their equivalent of photosynthesis. This doesn't mean they can't produce their equivalent of animals! It means that their more complex life forms are the ones that are naturally selected by their survival in a different atmosphere. To claim that life requires our specific biochemistry is simply argument by lack of imagination, or from personal incredulity. We never could have dreamed up our own biochemistry; we learned of it through observation. That is hardly proof that there could be no other.
Human history is incredibly young. As I understand it, in Star Trek, they constructed the Prime Directive as an explanation for why, if sentient beings are whizzing about the galaxy, they never had occasion to visit Sol, and Terra, before the space age. I don't know if that's needed. We might just never have been found. Hundreds of billions of stars might take some time to sift through. Or there might be some sort of Prime Directive out there that only applies to biological and not to cultural considerations. Our prehuman ancestors, anywhere from bacteria to primates, might have been observed from afar by beings who did not want to interfere because they could not calculate what impact their visitation could have on our climate or geology, and therefore our evolution, for example. We might think we're alone merely because They are cosmological non-interventionist environmentalists! Or maybe we're one of the most advanced life forms in the galaxy, but this will not necessarily be true of the intelligent life on our planet in a billion years, if there still is any. Contact could be a near certainty on a cosmological time scale, while still being incredibly unlikely within, say, the next ten thousand years, and still being merely improbable within the next ten million.
I'm not an exobiological nihilist, that is to say, I'm not saying we can't really know anything, that any of our hypotheses have just come from lack of imagination of alternatives. I think that the idea that life is common, but rare enough that isolation is also common, is probably true. But I would advocate more flexible thinking. I find the idea of the anthropic principle applied to our era to be compelling, and also that other, incredible, unfathomable biochemistries are possible.