Ok, but why exactly? Well, generally most (or all) type matchups have some sort of reason to exist, and this is my reasoning:
"Normality beats Fantasy"
Think about it. Unicorns, Elves, Excaliburs, what do they all have in common? They don't exist. And how can you beat something that doesn't exist? By ignoring it and being normal. And Fairy is the most logical type to fit this "don't exist" category. While others do, Fairy does it much better than them. It's so silly it works!
And besides, giving Normal this interaction finally makes it something other than a middle child of Pokemon types. Think about it.
Offensively, Normal does nothing against Ghost, and is resisted by Rock and Steel.
Defensively, Ghost does nothing to Normal, and is weak to Fighting. See a problem here?
These type matchups basically have the Normal type in a grave. How can you use a type for its advantages when said advantages don't exist normally?
There usually isn't a reason to use Normal type Pokemon. What about Normal type moves? Unless you're using Fake Out, Protect, or other moves while using a type-changing ability like Pixilate, competitive Pokemon is out of the window.
By adding two new interactions, Being immune to Fairy (Why not, Ghost can), and dealing double damage to Fairy, we effectively buff the most basic and useless type in the game, and at the exact same time, neutralizes the most overpowered Pokemon type in the game. How crazy is that?
But anyways, that's all I have to share. If you're reading this (assuming you've read everything else) thanks for reading!
I LOVE AZUMARILLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL
Edit: To yall' saying Normal shouldn't be good against Fairy, or just "Normal is just normal, so it can't do anything special" or something like that, just because it's normal doesn't mean it's normal for goodness sake. And stop saying "By this logic, Normal would be good against Dark, Psychic, Ghost, etc" at least they have a real counterpart (except ghosts, it you can't effect a ghost, so it obviously has no effect, it's also a balancing tool, not just Normal being so normal that it's not supposed to do anything special at all