Scope: You don't normally need to do this when changing stems, it is only necessary with new forks or if somehow the star nut has gotten crooked. A "star nut" is a thing that goes down inside the fork, and into which the top cap bolt threads for preload.

He starts the video by reviewing that Clearance of the bearings is adjusted by the top cap with the stem loose. The top cap screws into the Star Nut or Compression Plug. This video will review how to install the star nut or compression plug.

When do I need to do that? If it is stripped or pulled out.

 - shows it is convex
 - uses special tool (star nut setter) to screw eg: TNS-1 Star Nut Setter
 Tool is necessary to keep it straight, and get correct depth.

normally the star nut just stays in the fork if you change the stem?

Rarely necessary, but it happens if:

  It was installed crooked.
  It was driven too deep.
  The threads get damaged.
  The fork steerer is replaced.

Usually the old one is left inside and a new star nut installed above it, because removing them is difficult.

Important exception: If the fork has a carbon steerer, you do not use a star nut. Instead you use an expansion plug, which grips the inside of the fork.