exactly. with the intake air and fuel being denser, you can pack more in the same amount of space and get more power. Therefore you make the same horsepower at lower RPM compared to the "hotter" mixture. Making more power (or the same horsepower at lower RPM) then takes less RPM to get the same power, hence the fuel saving (slight fuel saving).
Another way to think about it is when you spark (burn) the mixture, it burns at the same final temperature at the end of the burn cycle (no matter where it initially starts burning). Now with the cooler air/fuel mixture is starting at a lower temperature. So by starting at a lower temperature before spark, then ending up at the same temperature when you finish burning, you are getting more heat out of the burn than starting with a "hotter" mixture (by about 20° to 30°). It is the change in the heat output that makes the available power. So you are getting another 20° to 30° of heat out of that charge. It is that heat that is released when burning that causes the expansion of the air fuel mixture on the power cycle to push down on the piston. The more heat that is released, the more power that the engine makes. This is purely a thermodynamic cycle (the otto cycle). The change in temperature and pressure is where the power is coming from.
