First of all, a lot is owed to the stairs (or stack of books, or shelves) themselves.
If the stairs you used were taller, the Slinky would flip over and end up rolling the rest of the way down. If the stair were shorter, the initial fall off the first step wouldn’t give the toy enough energy to continue its cool walk down.
That said, the simple answer is that when a Slinky is poised on the top step and you gently nudge the top portion over the brink, you place energy into the coil that transfers, coil by coil, down the length of the wire.
That energy moves in waves, like vibrating ripples, as the toy is pulled by gravity. The momentum of the toy’s movement bounces it over and down onto the next step, and the “walking” continues.