In practice it would work like this:
Lets say you have a puppet with a "Right Brow" that is driven by your right eyebrow. In addition to this, there is a "Right Wrinkle" above it i'd like to move as well, but not as much. So i'd drag the pick whip from "Right Wrinkle" to the "Right Brow" and then set the influence to say 50%. So if the brow moved 30 pixels, the wrinkle would move 15. I could have another "Right Wrinkle Small" that I wanted to move even less so i'd pick whip that to "Right Brow" as well and set the influence to 25%, and it would move only 7.5 pixels or what have you.
This would add a lot of subtle movement making animated creatures more natural. You could also use this for a character with multiple eyes or what have you.