Why did moving the CG aft on the Wright Brother's plane improve handling?
It did not improve stability, but flyability.

The Wright Flyer had the CG located aft of the neutral point and was unstable in pitch. A longitudinally unstable aircraft will deviate from its trim point, and when flown by a human pilot, will have a wavy flight path. The Wrights called this undulation. It is caused by the delay in the pilot's response and his tendency to overcompensate, so the aircraft is oscillating around the trim point (= the angle of attack where all moments around the pitch axis are in balance). This delay is more problematic when the eigenfrequency of the motion is higher. All Zeppelins were unstable in pitch above a certain airspeed and in yaw over the full speed range, but nobody bothered. These things were so big that any motion needed a long time to develop and deviations from the trim condition were easy to correct.

With aircraft, things are different, because they are more maneuverable and much smaller. The characteristic parameter here is the short period mode which characterizes the pitch motion. Its frequency is around 1 Hz for small aircraft and can be approximated like this:
$$\omega_{\alpha} = \frac{v_{\mathrm{trim}}}{i_y} \cdot \sqrt{\left(\frac{c_{m\alpha}}{\mu} + \frac{c_{L\alpha}\cdot c_{mq}}{\mu^2}\right)}$$

Nomenclature:
    $\omega_{\alpha} \:\:\:$ Eigenfrequency of the short period mode
    $v_{\mathrm{trim}}\:$  trimmed airspeed
    $i_y \:\:\:\:\:$  Radius of inertia around the pitch axis
    $c_{m\alpha} \:\:$ pitch moment gradient over angle of attack
    $\mu \:\:\:\:\:\:$  reduced mass. $\mu = \frac{2\cdot m}{\rho \cdot S \cdot l_{\mu}}$
    $c_{L\alpha} \:\:$ lift coefficient gradient over angle of attack
    $c_{mq} \:\:$ pitch damping coefficient
    $m \:\:\:\:\:$  mass
    $\rho \:\:\:\:\:\:$  density of air
    $S \:\:\:\:\:\:$  reference area (normally wing area)
    $l_{\mu} \:\:\:\:\:$  mean aerodynamic chord

To answer the question, all it needs is the first factor of the equation. The frequency increases with airspeed and goes down with the pitch inertia of the plane. By adding a weight at the back of their airplane, the Wrights increased this pitch inertia, thus lowering the eigenfrequency and making it easier to control. Make no mistake: The Flyer became even more unstable, but the instability became easier to correct because the deviations from the trim point built up more slowly.

Modern replicas of the Flyer have their CG ahead of the neutral point and are naturally stable.