The explanation isn’t quite right, it is true that an aft CG is more fuel efficient but it is not the prime concern when establishing the CG limits. The CG limits are based on stability and controllability, too far forward and the aircraft becomes too stable, requires high stick forces to manoeuver and at some point dives uncontrollably as the horizontal stabilizer isn’t effective enough to compensate for the nose down pitching moment. Too far aft and the longitudinal stability is insufficient, theoretically also increasing fuel consumption as more control inputs are required to maintain the correct flightpath. Under certification standards an aircraft needs positive static stability which is the determining factor for the aft CG limit. So an aft CG decreases fuel consumption only up to a certain point after which it starts increasing again.
The reason why an aft CG causes lower fuel consumption: in normal flight, on a conventional aircraft the CG is located in front of the wing center of pressure and the aircraft therefore has a natural nose down tendency, the further aft (therefore closer to the CP) the CG is, the smaller the nose down pitching moment becomes (and vice versa). The nose down moment has to be compensated for by the horizontal stabilizer + elevator, this causes a “downforce” on the horizontal stabilizer which basically is negative lift or can even be considered as weight. The wing has to compensate for this stabilizer downforce by creating more lift and therefore, all other factors being the same, a higher AOA is required for straight and level flight for a forward CG as compared to an aft CG.
Result: more lift required means more induced drag, the increased deflection of the elevator and downforce cause an increase in trim drag, the increase in drag (induced + trim) causes higher fuel consumption, lower range and endurance, lower climb rate. The required increase in AOA causes an increase in stall speed.