When bike racers talk about their bodies, we talk about our legs, our butt, our feet. That makes sense - in achieve maximum speed it's crucial that all the power coming out of our legs goes straight into the pedals. And we sit on our butt most of the time (and it has power that goes into the pedals too).
So how about the hands, then?
Sometimes I realize that my hands just feel good on the hoods. They're just gripping them in a way that feels like they're a part of the bike. Sometimes they get numb on longer rides. Sometimes after a crash I think about them ALL the time, because every time I reach into my pocket for my keys or anything else they hurt. But I rarely think about them otherwise.
WHY HANDS ARE IMPORTANT FOR BIKE RACING
I think I learned the importance of hands for stability the hard way. Several years ago on the second lap of the amateur qualifiers for Twilight in Athens my front wheel was taken out while we were going downhill, I hit the deck and broke my collarbone. Thinking back on how I rode then, I was pretty relaxed and confident. Unfortunately that also meant a more relaxed grip on my handlebars. I don't know if I could have stayed upright with my front wheel getting knocked to the side like that by having a firmer grip, but I think I would've had a better chance. Now I always try to be conscious of having a firm grip on my handlebars, particularly when things get a little tense in the peloton.
Sometimes when the going gets tough, racers have a tendency to move their upper bodies around. Sometimes a lot. Bobbing shoulders, swaying side to side, any number of movements. I personally think that's okayish (though for the record I don't do it) IF you are very well anchored to your handlebars and seat. If you don't have a firm grip, then you're just losing power into the air. Generally speaking though, I believe having a firm grip on the handlebars improves power transfer. And in a sprint it's downright crucial.
Most of your steering comes from your body (you can turn the bike without having your hands on the handlebars), but when things are getting fast and angry, having a firm grip on the handlebars means you can move confidently and stay upright. If someone is bumping against you, BANG you can bump them right back. If you are well anchored to your bike, you are somehow a little more stable and hard to make move.
This is obvious, but you need your hands to shift gears, brake, get food our of your pockets, grab your bidon, signal, push someone out of the way, wave hello, grab a handup - basically everything other than pedaling and sitting (the big job of the butt - worth its own write up). If you appreciate your hands for doing all these jobs for you (in the same way you say "thanks, legs" for a burst of speed), they will be more likely to work well for you.
And that's just science. Just like a clean bike is a faster bike.