What exactly causes the stiffness from a rotator cuff injury? muscle spasms?

Pain, weakness, etc: Stiffness can be a sign or symptom. If truly stiff/loss of motion, then the diagnosis could actually be a frozen shoulder or arthritis. If just perceived stiffness, then pain, inflammation or weakness from a tear can give you the perception of stiffness.

Capsular tightness: A rotator cuff tear is usually painful and can spawn a cascade of undesirable events in the surrounding tissues. The muscles attached to the torn cuff tendon can become weak, atrophied and painful. The underlying shoulder capsule can become transiently stiff and noncompliant resulting in a debilitating loss of motion ("frozen shoulder"). Scar tissue/synovitis can accumulate and block motion too.