Lets give a general constraint and say that there are n number of people with blue eyes...
When n=1, that person knows it must be him as he sees no-one else with blue eyes. He leaves on the first night.
When n=2, each of the two blue eyed people see only one other person with blue eyes but, for all they know, that one person could be the only one with blue eyes. The other person sees the same thing. BUT, when after the first night neither of them have left, each one deduces that the other must be seeing someone else with blue eyes and, since there is no-one else to be seen, that person must be them. They both leave on the second night.
When n=3, each person sees two others with blue eyes and expects them to leave on the second night, for he (being a perfect logician) has already considered the previous possibility. When after the second night, no-one leaves, he realizes that he must have blue eyes too. Three people leave on the third night.
And so it goes on... All 100 blue-eyed people count 99 others and know that if, on the 99th night, no-one leaves then they must have blue eyes.