What does it mean to have a strained rotator cuff?

Not as bad as a tear. The tendons that attach the rotor cuff muscles to the top of the shoulder bone are commonly injured in adults. The spectrum of injury runs from a mild strain to a partial tear and ends with a full thickness tear. A strain specifically means that no actual tendon fibers are torn or ruptured, whereas increasingly more tendon fibers are overtly torn with partial and full thickness tears.
No much. A strain implies the muscle has been overloaded in some fashion but no structural damage has occurred. The fact that it hurts implies injury but not all injury is structurally significant.