Yes, level boosters in Guild Wars 2 generally skip the story. They are mainly designed to quickly boost your character's level, so you can jump ahead in terms of character progression without having to go through the story content that would normally be associated with leveling up in the traditional way.
Level boosters in Guild Wars 2 are intended to accelerate your character's leveling. As a result, they do skip the story. The story in the game is an integral part of the overall experience, as it provides context and background for the game world and your character's place in it. If you use a level booster, you'll find yourself at a higher level but without the story knowledge that comes with gradually leveling up. You can always go back and experience the story later, but it won't be the same as experiencing it as you level up.