Bowser has kidnapped Princess Peach once again, but this time, Mario is tired of jumping around the kingdom to save her. So he instead takes a route that's a bit easier on his legs.
In this movie, the aim is to complete the game with the fewest A button presses possible. This movie uses more creative tricks and exploits than what you'd see in other movies for this game. A more in-depth explanation can be found in the authors' comments.
The total amount of A button presses done in this movie is 62, you can turn on Closed Captions in the youtube video to view a button counter.
You can watch Kosmic give an in-depth look of the run here as well as watching the premiere of the TAS here.
If you prefer console verification with no commentary, click here to watch that.