Basically, the player uses a glitch to get outside of the level data of the game's ROM code, and get into the other data. From there, they are able to "edit" the game code into thinking the player has beaten the game, essentially beating the game by only playing level 1.
More from the YT description:
"This TAS aims to complete the game visiting only the first level. In other words, the objective is to break the block that represents memory address 0xA2D5 just like the 2:09 run, but by going there directly. I was thinking of ideas to improve the 2:09 run but they all failed or are too slow.
The pause glitch itself has the potential to trigger the credits, due to its tendency to glitch the game up one way or another, but I'm not very technical so I can only keep testing until I get lucky. Andymac said he would try it if VBA had a trace log or some such feature."