Since you already understand the drag curve, consider next that in unaccelerated flight lift = weight and thrust = drag. So, at a given weight, lift is constant in level flight, and thrust required is directly proportional to drag.
So, with constant lift, L/D max is where the D is smallest -- that same point on the bottom of the drag vs airspeed curve. Minimum thrust is required at that point, therefore minimum fuel flow and maximum endurance.
What you apparently misunderstand is that max endurance IS at L/D max. However, max range is at an airspeed and power setting above the max endurance airspeed. Max range is a function of power required, not thrust.