Why would my hand be swelling after a rotator cuff surgery?

Surgery: This is not uncommon after rotator cuff surgery because fluids are typically used and after surgery that fluid collects and then drains along the path that gravity allows (down the arm to the hand). Also, the arm is typically placed in a sling that limits motion of the shoulder and elbow and thus collected fluid isn't readily pumped back upwards. Elevation, ice and hand and wrist motion can help.

Elbow bent in brace: Assuming you had an arthroscopic repair, a significant amount of fluid passes through the shoulder, and some moves in to the surrounding soft tissues creating the look of a big swollen shoulder. The mechanical effect of this fluid pressure together with the fact that the elbow is held flexed in a brace create a situation of lower than usual venous outflow manifesting clinically as a swollen hand.