When studied at ATPL level the conventional explanations for best endurance and best range speeds are as follows.
Endurance is the amount of time that we can fly using the quantity of fuel that we have on board the aircraft. So to maximise endurance we must minimise the rate at which we use the fuel. So maximum endurance in any aircraft type is achieved by flying at the speed at which the rate of fuel flow is minimum.
Fuel flow in a jet aircraft is proportional to thrust, so maximum jet endurance is achieved when the thrust required for a given TAS is at a minimum. In straight and level flight, thrust = drag, so we could say that the shape of the drag / TAS curve is representative of fuel flow / TAS curve. Best endurance occurs at the speed at which fuel flow is minimum, so in straight and level flight, the best jet aircraft endurance will be achieved at minimum drag speed, Vmd. Vmd occurs at the lowest point on the drag /TAS curve.
The relationship between fuel flow and thrust is less direct for a propeller aircraft. This is because the engine does not produce thrust directly. The engine produces Power through its output shaft. This power is fed to the propeller where it is used to generate thrust. The efficiency with which the propeller uses the power to produce thrust is not constant, but varies with such things as blade angle, RPM and aircraft TAS.
Fuel flow in a propeller aircraft is proportional to the amount of power that is being produced by the engine. So we can say that the shape of the power required / TAS curve is representative of the Fuel Flow / TAS curve. So maximum propeller aircraft endurance is achieved by flying at the speed at which power required is a minimum, Vmp. Vmp occurs at the lowest point on the Power required. TAS curve.
Note that in both cases maximum endurance occurs at the lowest point on the curve.
To achieve maximum range it is not sufficient to burn the fuel slowly. We must fly as far as possible for each unit of fuel consumed. This means that we will achieve maximum range by flying at the speed at which the ratio of Ground speed to TAS is a maximum. If we assume still air conditions this means that the ratio of TAS to Fuel Flow must be maximised.
For a propeller aircraft we look at the power required (or fuel flow) / TAS curve. To find the speed at which the ratio of TAS to Fuel Flow is maximum we draw a tangent from the origin to touch the curves. The maximum range speed is the point at which this tangent touches the curve. For all aircraft types, A tangent from the origin to the Power Required / TAS curve touches the curve at Vmd. So maximum range speed for a propeller aircraft is Vmd.
For a jet aircraft we draw a tangent from the origin to the Drag (or fuel flow) / TAS curve. This tangent touches the curve at about 1.32 Vmd. So maximum range speed for a jet aircraft is about 1.32 Vmd.
Note that in both cases (props and jets) max endurance is at the bottom of the curve and max range is where a tangent from the origin touches the curve.
BUT
The above explanations are based on the assumptions that the Power Specific Fuel Consumption (for propeller aircraft) and the Thrust Specific Fuel Consumption (for jet aircraft) are constant at all speeds. Because of these assumptions, the predicted speeds for best endurance and best range (Vmp and Vmd for props and Vmd and 1.32 Vmd for jets) are not entirely accurate.
If we wish to study the subject at a higher lever, such as for an Aerospace Engineering Degree, we would need to look into the aerodynamics as PBL has done in his posts. The conclusions would then be in a different format, and may look more impressive, but would still produce essentially the same results. If we limited our examination to aerodynamic factors and ignored the variable nature of SFC, we would again produce results that were not entirely accurate.