What causes the wave velocity of longitudinal waves in a spring to changes as its length changes? This video shows the velocity of longitudinal waves changing in a slinky spring as its length changes:
https://youtu.be/y7qS6SyyrFU?t=17
Something else responsible for the change in velocity correlated with the change in length is responsible; but what is it?
 A: The speed of waves in any medium can qualitatively be broken down to depend on two things:
1) The restoring "force" that tries to bring the medium back to equilibrium ($F$)
2) The "inertia" of the medium, or how hard it is to restore the medium to equilibrium. ($\rho$)
And typically the equation for the speed of the wave in the medium then has the form of
$$v=\sqrt{\frac{F}{\rho}}$$
Stretching the spring puts it under more tension. This therefore influences the first point above. The larger the tension, the larger the restoring force, and thus the faster the wave will be. Stretching the spring also reduces it's linear mass density. This relates to the second point above. This makes it easier for the larger force to pull the spring back to equilibrium, this aiding in an increased wave speed (the video description agrees with this analysis as well).

An analogy for the first point can be seen with a guitar string. Increasing the tension in the string causes the speed of waves on the string to increase. Since the wavelengths of the standing waves remains constant, this results in a larger frequency of oscillations in the string, which we percieve as a higher pitch when the string vibrates the air. Of course the waves on the string are transverse rather than the longitudinal waves in the video, but I thought I would bring up something more familiar.
