improved volumetric efficiency. Added power at any given fuel load reduces the work the engine is doing. pumping more air into the cylinder means a bigger bang when the fuel in injected, without using any more fuel. if you use less fuel for a given power output, it increases fuel economy. Reason being when you heat air it expnands. more air, more expansion, more pressure on the piston.
a turbo basically captures the energy of the hot exhaust gasses, and in a roundabout way returns that lost energy into the engine via compressed intake charge. again, more efficiency in the engine. more efficiency better mileage.