Any ideas why your face sags if you've had a stroke?

Muscle weakness. If the stroke affects the nerve that supplies the muscles of the face, these muscles are unable to contract and become lax, just as the ones in a stroke-affected arm or leg might become. It is this laxity of facial muscles that causes the sagging of the face.
Facial muscles. The stroke may affect unilateral arm and leg strength, and could also affect function over the face, causing poor facial nerve function of a central type, which causes the lower face to sag secondary to weakness.