The fundamental reason is that a jet engine's fuel consumption is proportional to the thrust produced (whereas a piston engine's fuel consumption is proportional to the power developed).
Therefore, the lower the thrust, the lower the fuel consumption.
As we all know, Thrust is equal to Drag, therefore if we fly at minimum drag speed (IAS) we will have minimum thrust and minimum fuel flow, which will give us maximum endurance (max hours per pound). We will be at the 'bottom of the drag curve'.
At high altitudes this IAS will result in a higher TAS, giving us more miles per pound.
To get maximum range, we must fly faster (about 1.35 x faster, as posted previously) than minimum drag speed to get a maximum ratio of speed/drag. We will then be flying at the 'tangent to the drag curve'.
So, in a nutshell, jet aircraft are more efficient at high altitude because the lower IAS associated with the high TAS results in lower drag which in turn requires less thrust (and therefore fuel flow) to balance it.