It's really very simple:
Instead of all the servers dumping cookies on you, you send a session-id to them, for instance 127 random bits.
In front of those you send a zero bit, if you are fine with the server tracking you, and you save the random number so you send the same one every time you talk to that server. This works just like a cookie.
If you feel like you want a new session, you can pick a new number and send that instead, and the server will treat that as a new (or just different!) session.
If instead you send a one bit in front of the 127 random bits, you tell the server that once you consider this "session" over, and that you do not want them to track you.
Of course this can be abused, but not nearly as much as cookies are abused today.
But it has the very important property, that all requests get a single fixed size field to replace all the cookies we drag across the net these days.