How does exercise help build and maintain strong bones and prevent falls?

Many ways. Studies show that weight bearing exercising like walking help with maintaining bone density. Exercises that do not have some weight bearing do not do as well with bone density. In general people who remain physically fit generally have better balance however certain exercises like tai chi and yoga have been shown to help with balance.
"Bones" and "Pones" Axial loading stimulates the bone forming cells to be active and add calcium to the framework of bone. Thus stronger mones. Lay around and the minerals leave the bones. Stay active, exercise and strengthen your muscles, especially your "core", and you will have better balance, gait and tone to prevent you from falling. This is more important the older we get. If >50---get with it!