Let's say you are holding the slinky and it isn't falling. Why isn't it falling? The same reason the sky isn't falling.
The atmosphere has layers of air and the layer below it has more pressure than the layer above it so there is a net force on the air pushing it up that is just enough so it stays there.
Same with the slinky. Let's talk a out the slinky while it is being held. It has layers and the whole thing is stretched more than it would be in deep space. That bottom most layer feels a force of gravity down and because the slinky is stretched it feels a force upwards (a stretched slinky has its parts exerting forces on each other that are pulling it together). It kept stretching until those forces balanced. So they are balanced. Now look at top layer it feels gravity down but you holding it up but you are holding it up more than the gravity pulling just that layer down you are holding with the force of gravity on the whole slinky. Bit since the slinky is strechted there is a third force pulling that layer down. The force pulling it down it equal to the Mg where M is the mass of the entire rest of the slinky (because that is equal and opposite the force that the rest of the slinky feels from the top layer).
So here are the forces on a layer. Each layer of mass $dm$ feels a force of gravity downwards of $gdm.$ Each layer is pulling up the layer below it with a force of $gM_b$ (where $M_B$ is the slinky mass below). And so it feels an equal and opposite force from the lower part of the slinky pushing it up with a force of $gM_b.$ Each layer is feeling a force upwards of $g(dm+M_b)$ (note that is the force of the mass below for the layer above since it feels that newest layer too, so it is all consistent). And the top layer feels that same force $g(dm+M_b)$ upwards except it feels it from your hand not from more slinky.
So the forces are $g(dm+M_b)$ upwards from upwards slinky layer or from hand. And $gdm$ downwards from gravity. And $gM_B$ from the layer below if any ($M_b=0$ for the bottom layer. So the. Net force on each piece is zero. Each layer feels its own gravity and feels forces from the layer above (or you have) and from the layer below (if there is one) but it all balanced out.
Now the forces from the different layers are caused by the stretching, it naturally is a certain length in deep space, each layer stretched until everything balanced out like above. But if you hit a slinky on the table and compressed it then after you stop hitting that one end then a ripple would travel through it at a finite speed, just like a wave wave does. That is based on the speed at which the slinky can make coordinated responses to changes.
So when you let go that top layer stops feeling a force up from your hand and still feels its own force down from gravity and still feels that force of $gM_b$ from the slinky layer below it. So it feels a force that would accelerate it much more than just g however once it starts to move it becomes less stretched so starts to feel less force pulling it down. This is the wave that starts at the top. It travels at a finite speed. But it takes time to get to the bottom. Meanwhile the bottom is still stretched so all those layers near the bottom are just as stretched as they always were and so exert all the forces they always did on each other so each layer still gets the force it needs to counter gravity.
Until that wave of changes starting at the top gets to it that is.