When you "sign out" you write your name and time you are leaving in a book (next to your name and time that you signed in) then leave. Visitors to a company often have to do this.
You could also "sign out" a book from a library in a similar manner.
"Sign off" is usually what a TV or radio announcer does when they are finished their story...when they say good-bye. It isn't written.
- For Teachers