While the given explanations make sense thinking about Piston driven engines... I am still having trouble understanding the concept with jet engines.
The kind of propulsion is actually not part of the problem. And therefore I think that the easiest way to understand it is just using a... sailplane 🙂
A sailplane basically loses altitude in order to overcome aerodynamic drag: it continuously trades potential energy (thanks to the loss in altitude) with kinetic energy (which is being dissipated due to drag). When is this exchange maximised? This exchange is obviously maximised when the sailplane loses as little kinetic energy as possible because of the drag i.e. when the drag is at its minimum (which happens to be also the point for best L/D). The relevant speed is termed "speed for best glide ratio".
Now, what happens when the sailplane reaches a thermal? Do we still need to minimise drag also in this case? No, the goal now is not only to gain as much altitude as possible but, since we don't know for how long the thermal is going to exist, the goal is also to do it in the quickest way as possible: now not only the travelled space (aka altitude aka potential energy) is important but the needed time as well; what has to be optimised now is therefore their ratio i.e. power by definition: the power extracted from the thermal.
So, if you want to:
- fly as much space as possible, you must fly at the speed for minimum drag (best L/D).
- fly as much time as possible, you must fly at the speed for minimum power required.
As said, note that this is a very general result not related to the particular propulsive system or aerodynamic characteristics of the airplane and therefore it applies equally well to an airplane with engine(s): obviously minimising the needed power minimises in this case the fuel consumption since that is the source of the power.
Due to the particular shape of the power required (as given for example in the plot in your question) the speed for minimum power is lower than the speed for minimum drag ($1/\sqrt{3}=0.76$ smaller to be precise) and the relevant drag is some 15.5% bigger.
