The Guru lines everyone up before speaking, so that all the blue eyed people are facing one direction, and all the brown eyed people are facing the other, and they alternate. So people with blue eyes can only see people with brown eyes, and people with brown eyes can only see people with blue eyes. The guru is looking in the same direction as the people with brown eyes are looking, so she can only see people with blue eyes.
When the Guru says she can see someone with blue eyes, all the people with brown eyes will know that she has oriented people a certain way, and will look at each other and see that all they can see are people with brown eyes, will know they have brown eyes, and will leave, as will the person the Guru was talking directly to. The next day, the remaining 99 people will look at each other and notice that they all have blue eyes, and they will leave. The Guru will remain forever.