Why cast arm for broken finger?

Depends on the break: Healing fractures need to be immobilzed. That means the joint above to the joint below. Sometimes the cast is extended to stop muscle activity as well.

Fractures in fingers: May need additional wrsit support to fully immobilize that fracture. The tendons that move the finger originate in the forearm, above the wrist toward the elbow. Wrist motion can cause tendon motion and in turn finger motion. Much depends on the type, location, pattern, stability, direction of the fracture. It may seem like too much but there is often a good reason.