How in the hell is this possible? Well, before letting it go, the Slinky is all stretched out because its weight is dispersed along its length. The top has the whole mass of the Slinky underneath it, and so the strain pulling down equals the weight, whereas the bottom has no mass underneath it, and so has no force acting on it. So the Slinky's spring tension winds up pulling up on the bottom at the exact rate it should be falling. When the top part of the Slinky catches up with the stationary lower part, though, gravity finally wakes up and slaps its stair-walking ass back down to Earth where it belongs.
Here's a giant slinky dropped from the top of a building, just to make it look freakier:
No one betrays the slink-mafia. No one.
This crazy floating act actually happens with any object you drop, because all solid objects have some degree of elasticity. It's just that the effect is much more noticeable with a Slinky because, you know, it's Slinky. For fun it's a wonderful toy.