It isn't even just the muscles in your hands. As I mentioned elsewhere in the thread, my wrist pain is caused by inflamed neck and shoulder muscles. Anything which builds torso strength can improve your posture and solve a lot of second- and third-order problems.
Plus, exercise helps muscles heal and stay healed. They like moving around. Exercise helps keep the blood and lymph flowing through muscles, both directly (I believe that moving around keeps fluids sloshing about and causes blood to be directed to the moving muscles) and indirectly (exercise builds cardiovascular health, which leads to better circulation and oxygenation). It even improves your sleep, which has many excellent side effects.
Plus, exercise helps muscles heal and stay healed. They like moving around. Exercise helps keep the blood and lymph flowing through muscles, both directly (I believe that moving around keeps fluids sloshing about and causes blood to be directed to the moving muscles) and indirectly (exercise builds cardiovascular health, which leads to better circulation and oxygenation). It even improves your sleep, which has many excellent side effects.