A major part of the reason we fly high, aerodynamic issues aside, is engine efficiency. A turbine engine is most efficient at higher rotational speeds; typically around 90% of the engine's 100% speed. At lower altitudes, the engine must be operated at lower power settings in order to meet speed restrictions (both regulatory, and airframe). The higher the aircraft goes,the more power is required,and we reach a point where the engine can be operated at optimum power; it's being operated at it's most efficient state.
This altitude depends on the amount of thrust required, which of course depends on the amount of drag the aircraft experiences...and this depends on the speed flown, weight, A0A, temperature, and other factors we consider when computing our optimum altitude. We're seeking an altitude where the combination of weight, speed, and thrust required occur where the engine is most efficiently operated.
If, for example, the engine is most efficient at 90% of it's maximum speed, we operate at an altitude where using 90% will give us the speed we desire, considering weight and temperature. As the airplane gets lighter, less thrust is required (meaning the engine is now operating at a lower, less efficient speed). We climb. As we climb, more thrust is again required, allowing the engine to be operated at it's most efficient speed.
The key to understanding that concept is understanding that the engine is most efficient at a given speed. Think of it like driving a car with a manual transmission up a hill (or a bicycle, if you will). Try using too high a gear and running the engine at too low a speed, the engine bogs down, the car bogs down, and the climb up the hill is inefficient. Instead, we climb the hill in a lower gear, and the engine runs at a higher speed, allowing it to produce the power necessary to get us up the hill. In a sense, the turbine engine is a little the same...it's most efficient at higher operating RPM's, and we achieve this at cruise speed at higher and higher altitudes as the airplane weighs less and less.