The truth is that your left hand is primarily the passive hand, while the right hand is responsible for the movement in your stroke. To understand how this works, try putting with your left hand in its normal position at the top of the handle but your right hand at the very bottom of the handle. To make a decent stroke from this position, you'll see immediately that your left hand must stay quiet while your right hand does almost all the work of taking the club back and then through the ball.
The advantage of the split-hand grip is that rather than trying to make your two hands work together in unison, each hand simply has one job to do: Your left hand is the anchor, while your right hand is the mover. This simplifies the stroke and makes it much easier to keep your putter on line.