Here was the way I saw it.
I am a blue- eyed person, but I can't see my own eyes, so I don't know that yet. I see 99 other blue-eyed people and 100 other brown-eyed people, along with one green-eyed guru, and I am a good logician. When the guru makes her proclamation that she sees one blue-eyed person, then I obviously know that someone has blue eyes, but who is it?
Suppose for a moment that I was the only blue-eyed person on the island. Then when the guru makes her proclamation, I know that I must have blue eyes, since all the other eyes that I see are brown or green; therefore, I must leave on the midnight ferry. Now suppose that I am one of two people on the island with blue eyes, so I see only one person with blue eyes. When this blue-eyed person does not get on the midnight ferry the night the guru makes her proclamation, I know that s/he must see another pair of blue eyes on the island. Since my eyes are the only ones I can't see, I know that they must be mine; therefore we both leave at midnight the next day. If you extend this logic to three blue-eyed people, they all get on the ferry the third day. If you extend this logic to 100 blue-eyed people, they all get on the ferry on the hundredth day.