How can a propeller behind a ship's hull improve propulsion efficiency?

Answer: It can recover some energy from water dragged along by the hull

It avoids the hull wake and works only in undisturbed water — A stern propeller normally works in the hull's wake, not outside it. The nonuniform flow can cause vibration, yet careful design can also exploit that wake to improve overall propulsion efficiency.

It balances the ship by moving all engine weight to the stern — Moving engine weight aft does not automatically balance a ship. Trim depends on the distribution of weight and buoyancy; the hydrodynamic benefit here comes from the propeller interacting with the hull's wake.

It can recover some energy from water dragged along by the hull ✓ — A stern propeller can recover some energy in the water dragged along by the hull. That water approaches the propeller more slowly relative to the ship, and the hull-propeller combination can need less power than propulsion in undisturbed water. The gain depends on the design, so stern placement is not a universal rule.