I think players think they do that...but in reality they don't.
At the same time I don't think any player ever said, "I think I'll just 'mail it in' on this AB". But for some players when they find themselves in a pressure situation they feel the tingle...the goose bumps of knowing THIS is where they want to be... and they can dig down and find a bit more focus than they usually bring. That's clutch.
Now.. are they always successful? No. Because baseball is such a random game that nothing is "ALWAYS". But these players are able to hit, field, pitch, or throw just a bit better than they usually do in those situations. It's reflected in their BA, etc. but since it's not identified as a pressure situation some folks think it doesn't exist.