How do I climb down from the hideout in Firewatch?

In the hideout, there is a table with a bunch of junk on it, including a placard of some description. If you pick up and move the placard, you will discover a set of pitons which are required to progress any further.

Like the rope and the flashlight, the pitons are an unlimited use item that let you create a place to hang some rope when there isn't a place to tie some rope otherwise. The only requirement is that you have some place to drive the piton into first.

At 1:27 in the video you linked, the player looks back at the hideout and zooms in. You can clearly see a highlighted crack in the rock:

That is where you were supposed to drive a piton into, creating a place to hang a rope and allowing you to rappel down to the path below.