With a catamaran you can approach the mooring slowly from downwind and hopefully downcurrent, and use differential thrust from the two engines to steer the boat to the mooring ball or anchor location.

This can be done at speeds below "steerageway" because you have two engines that are far apart. To turn to port, you give more forward thrust to the starboard engine, as done at the end of the video. And you can put one engine in reverse to turn faster to that engine's side.

For really small adjustments towards one side you can put that engine in neutral. With the engine on the other side still in forward, it will push the bow toward the side that's in neutral. That's also done in the video.

(Steerageway is the minimum speed through the water for the rudders to be effective. Below that speed, the rudders don't do much. That's true on both monohulls and multihulls, but it's why we use propwalk and propwash to turn the boat at low speeds (below steerageway) on a monohull.)

The video was correct in showing the helmsman doing this with the throttles, because picking up the mooring pendant is the same as a monohull, with the exception that the bowman is using a gaff (boat hook) presumably because the bridgedeck is too high above the water to reach the pendant by hand.

The pendant is attached to the long yellow pole on the small red buoy used to pick up the mooring line. It's attached to the mooring lines and thus mooring ball. Normally the pick up pole is supposed to be sticking up a couple meters straight out of the water.

All of this is a bit subtle to folks who haven't done it before, so some of the confusion in the comments is understandable.

The video is intended to transition experienced monohull crew to catamarans.