How does one move past this fear? First, focus on the process itself. For example, a hitter must focus only on what is under his control: the execution of his swing. "See ball, hit ball" are his only considerations when the pitch is in the air. Fear, worry and tension are prompted by thoughts of expectation: "I must get a double, I must get a great hit, I must not swing and miss." Nonsense. Hit the ball, and hit the ball hard, and everything else takes care of itself.
This is not a guarantee of success - even if you hit the ball well, a fielder can still make a great play. However, the act of focusing on your swing (the process) returns your consciousness to your actions rather than the observations of someone else (their expectations). This is what is meant by the cliche "play within yourself." If you do this, the only time you'll notice those watching, is when they cheer your latest and greatest hit!