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 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.