The explanation that "it takes time for the bottom of the slinky to feel the change" might work ok, but it isn't the best.
Then why doesn't the bottom of the slinky fall as the top is let go? I think the best thing is to think of the slinky as a system. When it is let get, the center of mass certainly accelerates downward (like any falling object). However, at the same time, the slinky (spring) is compressing to its relaxed length. This means that top and bottom are accelerating towards the center of mass of the slinky at the same time the center of mass is accelerating downward.
Physics blows my mind. When you watch this, you'll notice that the bottom of the slinky doesn't move until the top gets to it. Here's the reason for that:
Found on kottke.org.