From that page:
The first aspect of the paradox, “the argument by scale”, is a function of the raw numbers involved: there are an estimated 200–400 billion (2–4 ×1011) stars in the Milky Way and 70 sextillion (7×1022) in the visible universe. Even if intelligent life occurs on only a minuscule percentage of planets around these stars, there might still be a great number of civilizations extant in the Milky Way galaxy alone. This argument also assumes the mediocrity principle, which states that Earth is not special, but merely a typical planet, subject to the same laws, effects, and likely outcomes as any other world.
The second cornerstone of the Fermi paradox is a rejoinder to the argument by scale: given intelligent life’s ability to overcome scarcity, and its tendency to colonize new habitats, it seems likely that at least some civilizations would be technologically advanced, seek out new resources in space and then colonize first their own star system and subsequently the surrounding star systems. Since there is no conclusive or certifiable evidence on Earth or elsewhere in the known universe of other intelligent life after 13.7 billion years of the universe’s history, we have the conflict requiring a resolution. Some examples of which may be that intelligent life is rarer than we think, or that our assumptions about the general behavior of intelligent species are flawed.
Thinking historically about this fact, I think it goes even deeper that that. Humans have been biologically modern for at least 100,000 years. Yet only in the last 100 years have we developed the capability of space flight. This capability relies a series of causal chains that go right back to development of settled agriculture in the fertile crescent 10 to 5 kya. At any moment in our history the links in those chains might have been broken. For example, the ice-age didn’t end. The earth never had a large continent (Eurasia) orientated east-west for the efficient spread of agricultural technologies among human societies (i.e. such that there was a large area of similar longitudinal climate similarity, and thus agricultural methods could be communicated). What if Rome was defeated by Carthage in 216 B.C after the battle of Cannae? Then it would never have conquered Greece and the eastern Mediterranean. How different would European history have been? Would there even be a “Europe” to have the “European enlightenment”? And so on and so forth that all comes to a head in 20th century European history leading, in its middle period, to two large hemisphere-dominating empires with roughly equivalent technical capabilities willing to expend a superhuman amount of material wealth to dominate the Earth from space. And it was just plain dumb luck that on any one of several occasions we did not blow ourselves up several times over!
Taking that casual chain all the way back into deep history I don’t think it is in any way inevitable that our species would get this far, in that we could ask these sorts of questions about galactic life, and have the tools we do to explore the problem. For most of human history we’ve been pretty good at shaping flint into cutting, sawing, hacking, piercing, poking, slashing, and bashing tools. We’ve also been adept at telling stories around campfires and painting on rock surfaces. And that’s the greater part of our natural history.
I would expect that life in the Galaxy is abundant. I reckon there’s moons of Jupiter and Saturn that will turn out to have at least things like slime moulds. Complex life (e.g. basic animals, photosynthesising plants) is fairly common. Intelligent species, which we recognise as such, e.g. visible language, complex symbolic representation (art), tool use, etc, will number maybe in the hundreds of thousands. Star-voyaging species? Less than 1.