It means that you are not interested in or worried about something or someone; that you place no value on something, an event or a person; that you do not care and are uninterested. It means that you place less value on it than you do on a turd, says it all really.
“Malcolm had a hissy fit when he found out he wasn’t invited to your birthday bash.”
“I don’t give a shit.”
I don't give a shit is a popular phrase to be used when you are not interested in what someone else says or believes. It implies that you don't even care enough to physically give them a sample of your fecal matter.
Gamer 1#: Dude your computer sucks, you shell out $2000 for a kicass rig, and still run it on a piece of shit CRT monitor.
Gamer 2#:Well that may be but honestly, I don't give a shit.