The object of the big ball is to keep your arms and swing synchronized and in place during the golf swing.
Notice that the ball does not come into contact with the body. This is the result of the target arm not collapsing (or the elbow folding) as the swing moves to this position. If it did the ball would be pressing against the chest.
In order for the swing to remain strong and not controlled through the hands or elbows the triangle must remain intact throughout the golf swing.
If you move the chest and shoulders while squeezing the ball between your forearms during all phases of the golf swing it will maintain a positive triangle of shoulders, arms, hands pulling down and into the ball as the lower body and core are rotating.