C K Dexter Haven: That’s generally the right idea. If a random mutation happens to be better-suited for the environment, then that organism will survive better than its competitors. What counts as an environmental pressure can be pretty much anything and everything.
As for the question at hand: Given how huge the universe is, I’d imagine the evolution of intelligence has got to be extremely common. But you have to take into account what “common” means.
We use standard candles in astronomy all the time (they help us dictate distance to faraway objects), and good sources are exploding stars. However, the kind of stars we refer to maybe happen once every 100 years in a given galaxy. That seems absurdly rare. And yet if you were to zoom in on the spot formed by holding a dime up to the sky, there would be 100,000 galaxies or so; which means on any given night, you’ll see around 10 stars explode in just that one tiny section. This is what we see empirically.
Now when you consider how many “dimes” you could hold up to the sky, that number is huge. The point: “Rare” events happen all the time.
If you wish to intuit just how big our universe is: The Known Universe by AMNH - YouTube
Life formed on this planet because conditions were ripe for it. We have a very stable orbit and we had conditions that ultimately gave rise to abiogenetic events (self-replicating molecules) which formed cells and, over billions of years, multicellular life and eventually intelligent life.
We’re made of the most common stuff in the universe, too, with a direct one-to-one mapping (hydrogen, nitrogen, oxygen, carbon, etc). Given how many galaxies there are, and how many stars exist, and thus how many planets are out there, a huge number of those are going to have stable orbits. It’s hard to imagine abiogenetic events NOT occurring at least a few times on all those planets.
Intelligent life is another question. I have no idea how common it is but I think it’s probably very common. However, the dinosaurs seemed to have supreme reign over our planet until they all died out (a rather unfortunate environmental pressure on their end) and allowed for a “shock” away from that local optima so that other creatures could continue to evolve and, well, not get eaten/outclassed.