"The Fermi paradox is the conflict between an expectation of a high {\em ex ante} probability of intelligent life elsewhere in the universe and the apparently lifeless universe we in fact observe. The expectation that the universe should be teeming with intelligent life is linked to models like the Drake equation, which suggest that even if the probability of intelligent life developing at a given site is small, the sheer multitude of possible sites should nonetheless yield a large number of potentially observable civilizations. We show that this conflict arises from the use of Drake-like equations, which implicitly assume certainty regarding highly uncertain parameters. We examine these parameters, incorporating models of chemical and genetic transitions on paths to the origin of life, and show that extant scientific knowledge corresponds to uncertainties that span multiple orders of magnitude. This makes a stark difference. When the model is recast to represent realistic distributions of uncertainty, we find a substantial {\em ex ante} probability of there being no other intelligent life in our observable universe, and thus that there should be little surprise when we fail to detect any signs of it. This result dissolves the Fermi paradox, and in doing so removes any need to invoke speculative mechanisms by which civilizations would inevitably fail to have observable effects upon the universe."
Another explanation for the Fermi paradox I had heard, which certainly is less rigorous than the linked article but easier to understand is:
The root of the paradox is that with so many planets in the universe (in the order of 10^24), there must be more life in the universe. But even with this unbelievably large number of potential planets, if creation of intelligent life on a planet requires only 24 independent parameters to be favourable, and each of them are favourable with a probability of 10%, then we should expect intelligent life in just one planet: ours.
"The Fermi paradox is the conflict between an expectation of a high {\em ex ante} probability of intelligent life elsewhere in the universe and the apparently lifeless universe we in fact observe. The expectation that the universe should be teeming with intelligent life is linked to models like the Drake equation, which suggest that even if the probability of intelligent life developing at a given site is small, the sheer multitude of possible sites should nonetheless yield a large number of potentially observable civilizations. We show that this conflict arises from the use of Drake-like equations, which implicitly assume certainty regarding highly uncertain parameters. We examine these parameters, incorporating models of chemical and genetic transitions on paths to the origin of life, and show that extant scientific knowledge corresponds to uncertainties that span multiple orders of magnitude. This makes a stark difference. When the model is recast to represent realistic distributions of uncertainty, we find a substantial {\em ex ante} probability of there being no other intelligent life in our observable universe, and thus that there should be little surprise when we fail to detect any signs of it. This result dissolves the Fermi paradox, and in doing so removes any need to invoke speculative mechanisms by which civilizations would inevitably fail to have observable effects upon the universe."