It can be used when you don't know who to blame or when you don't want to confess.
This phrase was first made popular in the amirite.net community in August 2010.
Tia was a fake persona created by a member of the website to be used as a scapegoat, but she was soon being passed the blame for other incidents by other members too.
Karla: "... Tia did it!"
The term originally appeared in August 2010 on the amirite.net chat room when one user told a BS story that got everyone enraged and then blamed her actions on her made up sister "Tia".
Brock: Who was it then?
Tia Creator: My sister.
Brock: What's your sister's name?
Tia Creator: Tia.
Translation: Tia Did It
Jenny: "...Tia did it!"
Johnny: "What the fuck, Jenny? GTFO."