Instead, let go of the bars, wrap your fingers into a loose fist and rest the heels of the palms of your hands on top of the bars. Your fingernails should be tapping the top of the bars too. This way, all of your weight is resting on the bony part of your hand and won't be sore. If you find yourself bouncing around, relax and spin your legs more. Smooth form on the hills really help me go faster while keeping the HR down. Also helps keeps the legs from fatiguing on long climbs.
Lemond calls the motion "scraping mud off the bottom of your shoes". It helps people who are still pushing down on the pedal at the bottom of the crank-revolution modify their pedaling-motion to be pulling back at 90-degrees to the crank instead.
What happens when you push on the pedal at the bottom is that you're trying to stretch the crankarms and bend the pedal-axle. Not gonna happen, so instead, all that force pushes your upper body upwards and to the opposite side of the bike. Then people grip the bars and pull to counteract their upper-body moving upwards. You're basically negating the useless force from the legs with an opposing force from the back-muscles. Neither of which does anything to move you forwards on the bike, but it does work the back and and cause lower-back pain and soreness.