I win more games when I stop caring about the outcome

All about them mentality.

Could be for any number of reasons.

For me at least I stop focusing on what the team should be doing when I stop caring about winning and focus more on what I in particular should be doing. This makes me play better and lets me have a stronger impact on the match.

Less pressure, allowing you to relax and make more calculated decisions?

I can play pool fine until it's time to get that 8-Ball in. And by fine I mean not as atrocious.

Because now you are not playing to win, you are playing to get better...

Winning happens from getting better!