When you finish a telephone conversation, you "hang up" the phone. Originally, people used to actually hang the earpiece of the phone when they were finished. We still say "hang up" today, even though you're usually just pushing a button.
To "hang up on" someone means to suddenly stop the call while they are talking. People usually do this because they're angry. It's rude to hang up on someone.
In the example above, the speaker says that her sister "hung up on" her as a joke. Actually, the telephone probably got disconnected, or maybe her sister hung up accidentally.