How does a runner during a Any% speedrun skip essentially the entire game, it looks to me like they just jump and restart dialog but on my attempts nothing happens.

You have to press E 12 times before you press the space bar

so basically when you click 'e' to start the dialogue you have to press e an additional 11 times so 12 in total there has been tutorial if you want to watch

There's also an alternative "Super skip" run that isn't yet recognized by speedrun.com. It allows you to complete the "kill mr krabs in his final form" quest without having to do the boss fight. Hopefully they add it as a category in the future.