Why this inertial regulator of dragonfly wing pitch reduces self-excited coupled flapping and feathering vibrations and improves gliding flight speed
Short answer:
Adding a mass ahead of the elastic axis of the wing tends to increase the critical flutter speed of the wing, thus allowing a larger flight envelope. Above this critical speed the wing would experience vibrations of increasing amplitude that could lead to its destruction (or to a limit-cycle flutter mode, which is also bad but less so than rapid unplanned disassembly).
Flutter is a destructive aeroelastic phenomenon that can be described (in a simplified way) as a combination of two natural vibration modes of a wing. These modes are torsion around a spanwise axis and bending with respect to the root. Each mode will have a natural frequency, and the flutter speed will have a minimum (not guaranteed to be a global one, btw) when both of these frequencies are the same, because then they will reinforce one another.
The distance between the center of mass and the elastic axis of the wing is very important for the torsion response of the wing, simply because the wing will twist around this elastic axis, and a center of mass far away from it will give high inertial moments.
By adding weight to the leading edge, the dynamic behavior of the wing is modified in a way that moves the critical flutter speed upwards, extending the safe flight envelope. This technique can be seen most often in helicopter blades, many of which have some trim or balance weight at the leading edge.
Long answer (math ahead):
WIP - The analytical treatment is quite long so I am trying to decide which results are actually worth typing into MathJax