Yeah early shifiting does save a lot of fuel in GT Sport. At Brands Hatch GP for example, without doing it and just shifting near the rev limiter you'd run out of fuel in lap 9 with the GT-R GT3. By Doing short shifts however you'd have about 1/8th left near the end which you can use to have some more power to gain time or overtake.
You can see it in my video as well, I started doing it from lap 3 onwards when I noticed I was using too much fuel already. Was on fuel map 6 the whole race, which Marcolah probably used as well. He ran out of fuel in the last lap while I easily made it and could even do a faster last lap.
As for how and why, the easy comparison I make is driving on the streets in your road car. You shift at low RPM as well to save fuel. If you keep shifting at high RPM's you'll be visiting the gas station more often as well.
Makes sense to me anyway that more fuel is burned in the higher RPM's since that's where most of the power is at.