The monkey hovers in the air because its weight is balanced by the tension force of the slinky. It remains hovering as long as the bottom end of the slinky continues to be stretched by the same amount. From simple mechanics we can understand why the monkey does not drop immeditely, because it takes time before the unstretching progresses down the slinky. However, why must the entire slinky collapse before the bottom end starts to unstretch?
The above video shows that when a rubber band is used instead, it still takes sometime before the bottom end starts to unstretch. However, unlike the slinky, there is no need for the entire rubber band to collapse before the bottom end starts to unstretch. So what’s so special about the slinky?
I was about to delete the above video when I noticed something interesting. By accident, a longitudinal compression wave was sent down the slinky just before it was dropped. So this video captured two “things” racing down the slinky. There is an ordinary longitudinal compression wave that travels at a speed as dictated by the mass and tension of the slinky. Hot on its heels is the collapsed slinky. The video shows quite clearly that the collapsed slinky actually travels faster than an ordinary compression wave would down the slinky. The “collapse”, being accelerated by both gravity and the tension force of the slinky below it, pushes into the slinky faster than a compression wave can propagate down the slinky.
That explains why the monkey does not drop before the slinky collapses totally. The medium itself (the slinky) travels faster than the wave. The entire slinky collapses before the disturbance can reach the monkey.