That video is not a TAS, no, it was the previous WR done on console. Current WR can be seen here:
http://www.twitch.tv/kerahime/c/2415937.
For comparision, here is my TAS:
http://tasvideos.org/3936S.html. I have some info about the run there as well as linked to my old run that has more info.
As for the warp glitch itself. When you open up the bonus in Orang-utan Gang, you must have Donkey Kong in the lead and not be too close to the entrance. If you are all the way to the right, a certain memory address will get a different value and you'll be taken to a different location. To prevent this, open up the bonus a little bit earlier, around when seen in the videos.
Once inside the bonus, you have to destroy the DK barrel and exit with both Kongs on the same frame. Most real time runners use a setup to stand at a specific location, throw the barrel and then move to the right. You can try this setup as well or just time it and hope to get lucky =).
To avoid walking automatically on the map when you finish a level, simply hold Right.
I hope that helps you. If you have further questions, please ask and I'll do my best to answer =).