This option allows you to follow a thread so you get a notification when there is a new reply in the thread.
How does this work?
Information about the threads you are following is stored in your browser's memory and never leaves your machine.
Are my thread alerts lost when I close my browser?
The behavior is browser dependent, but in all common modern browsers it should persist between sessions (i.e. if you close your browser and then come back to the site in a new browser session).