You're right, my explanation is actually wrong. The idea is right, but the recursive knowledge is not gained incrementally the way I described but rather all at once at the end when they wake up on the nth day and see that everyone is still alive.
Each one reasons (recursively) that if their eyes are brown then, given that common knowledge of the existence of blue-eyed people has been established, the remaining n-1 blue-eyed people will realize their eyes are blue (and kill themselves) on the n-1'th day. Since this doesn't happen, their eyes must be blue.
Each one reasons (recursively) that if their eyes are brown then, given that common knowledge of the existence of blue-eyed people has been established, the remaining n-1 blue-eyed people will realize their eyes are blue (and kill themselves) on the n-1'th day. Since this doesn't happen, their eyes must be blue.