At lower altitude an aeroplane usually has more lift. However an aeroplane flying at low altitudes (with gear/flaps up) at low velocity burns the same amount of fuel it would flying much faster at a higher altitude. Why?
The lift basically depends on the the velocity of the plane and the the density of the air around it. That means, in higher altitude, a plane needs a higher velocity to maintain its height, BUT: The velocity is not just for staying up, but obviously is a desired feature of an airplane, thus, they would not speed down even it was possible while maintaining height.
On the other hand, the higher the pressure and thus the density of the surrounding air, the higher the drag. As we already find out that velocity is desired, the drag results in more energy needed to maintain the velocity. Thus, it is more efficient to fly at higher altitudes.
Airplanes have a pitot tube, that has an air hole facing forward and one facing sideways. The difference of pressure between these two is what drives the airspeed indicator.
At a higher altitude, the air is thinner, so to get the necessary lift the airplane has to travel faster through it. However, since the air is thinner, it takes higher airspeed to get the same pressure differential in the pitot tube.
The result is that the indicated airspeed needed to fly is the same regardless of altitude. Practically all of the speed-dependent behavior of the aircraft, including lift, drag, fuel usage, is keyed to indicated airspeed, not true airspeed.
So at higher altitude, since the true airspeed is higher, you can cover more ground, everything else being equal.
(Of course, there's more to it. For one thing, there's less oxygen for the engine, so less power. Also, the winds blow much faster the higher you go, so if your plane is not very fast, but your headwind is, you ain't goin' anywhere. I had that experience once. Took off, fly fly fly, looked down, and the airport was right underneath me.)