The idea behind this handle is that the extension "hooks" onto the fifth proximal phalanx of the hand, and can then be used to pull the racket without much "gripping" force (this means decontraction for everyone!). Gripping force during acceleration is in general not a big deal in shots where pace isn't the issue because a proper preparation allows using very little force during acceleration. But it can be a problem on very fast incoming balls, where time for preparation (especially in combination with ball seeking) is limited.
Once the racket head approaches the contact point, the extension is also meant to be used as a pivot, to help flick the racket head and whip the ball. This is what helped me finally develop what I feel is a decent topspin forehand that I'm happy with. But, most importantly, that is extremely consistent, and from which I can pull off spin on most shots.
Finally, right on and right after impact, this handle also reduces the amount of gripping force (rather, locking-onto-the-buttcap force) needed to prevent the racket from slipping, effectively reducing the amount of muscle tension right at impact, that reduces the vibrations from the shock that get transfered to the forearm and elbow.
One important clarification: this handle only improves and solidifies the forehand. The extension is not meant to be used on the backhand side at all. To avoid having the extension in the way on the backhand, it's enough to play the backhand as usual, and not with the same face as the forehand. In the serve the extension is also not meant to be used.