*Pitcher's Hand Explained

The pitcher's hand rule denotes the pitcher as a proxy for the (non existent) First Baseman on any force plays at 1B.

 

On any ground ball hit (to infield or outfield) the fielding player will attempt to throw the ball to the pitcher before the batter reaches 1B. If the pitcher catches the ball and clearly secures it before the batter touches 1B, the batter is out. This also includes when a force out has already occurred at another base resulting in a double play. 

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In Pitcher's Hand the pitcher may "travel" anywhere that he chooses, including as close to the fielding player as possible, before making a catch.  

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The pitcher is NOT a proxy for the First Baseman after a Fly Ball or Pop Up out has occurred. Example, with a runner on 1B and a Fly Ball to LF, the runner may float off the bag without worry that the LF only need to throw it to the pitcher before he gets back to the 1B bag resulting in a double play. In that scenario the pitcher would have to be touching the 1B bag to get the second out.