Most people convince themselves -- and will often feel the need to convince others as well -- that they don't really care because they are aware the outcome of a situation is not in their control.
No one likes feeling powerless, and to admit that something honestly matters to you means you are vulnerable to the possibility that you may not get the result that you actually want. The more important getting a particular result becomes, the more your life will seem to revolve around it. As long as you believe it doesn't matter, the outcome, good or bad, doesn't change your sense of control.
This is generally referred to as "sour grapes". It's a reference to the fable about a fox that saw some grapes in a tree, and instead of trying to get them, he avoided having to admit defeat by convincing himself the grapes were sour and that he didn't even want them anyway. He would rather have a guaranteed outcome that he chose than to get the exact same outcome despite his effort to reach an alternative.
It's easy to avoid feeling powerless by doing stupid things. For example, some people troll the internet anonymously with the intended goal of getting a negative reaction because it's easy to find someone who will be mildly upset enough to respond. The bigger the reaction is, the more fulfilling the task becomes. It doesn't even matter how clever your reply to a troll may be -- the fact that they weren't completely ignored satisfied their interest in that task.
Another example is when you play a multiplayer game. There's always that guy who needs to announce to EVERYONE early in the game that he's not even trying to win because he doesn't care. He needs you to know that, and will sometimes go as far as sabotaging his own team to prove it, so that if\when he loses, he can't be criticized for it. He's fine with losing as long as he's the one who makes that choice.
So how do you get over the "angry teenager" mentality and begin to give a fuck?
Accept that there are things you care about. Shit will happen and there's nothing you can do about it except move on. You'll feel powerless when things that are important don't go your way, and when that happens, that's the time when you need to remind yourself that while you weren't in control of that outcome, you are in control of how you to handle the disappointment.
Do you pout and act like a child when you're disappointed, or do you accept that you want to be better than all that?
Sometimes, you may not get the grapes either way, but at least if you admit you want them, they'll taste that much sweeter if you are able to reach them. If you want to be more assertive and have a defined set of priorities, you have to accept the risk of having things that matters to you.
tl;dr It's easy to say you don't care. It's difficult to admit that you do.