Fairies by their nature are gentle creatures, so the fairy typing is all about calmness. Fairy energy subdues and calms the types that are focused on combat and aggression above all else. This is why they're good against the honor-combat-oriented fighting types, the combat-pragmatism-based dark types, and dragons, which as a type is defined by moves like Outrage. All of these types are defined by aggression, offense, and combat, which is why fairies can so easily subdue them. The bug resist is because bugs are simple-minded creatures, and fairies are just so much grander in the scheme of things.
However, these fairies that are focused on tranquility and serenity are helpless against types that are physically destructive, like fire. The fairy can calm down the raging dragon but it cannot stop a fire so easily. For the same reason, they're completely helpless against disease; they can calm the spirit but cannot heal the body. Fairies are themselves especially susceptible to poison. While they're strong enough in spirit to ward off the "impure thoughts" from a dark pulse, this mental immune system comes at the cost of their actual immune system.
Okay this has been bothering me since Fairy as a type was first revealed. I'd like to point out that fairies are weak to Steel because Iron is a anathema to faeries. As is poison, for instance if a Dryad were to have it's tree poisoned it would be poisoned as well. hence why that is effective against it's type. It makes perfect sense.
As for being immune to Dragon...well take the Brownie for instance. They are small and generally cute looking but Brownies while generally helpful and adorable...can tear a grown man apart.
The Fairy type are immune to Dragons mainly for gameplay reasons but when you consider most fairies are much stronger then they appear to be it starts to make more sense. The Coliunn Gun Cheann (Headless Trunk) Was an enormous creature with no head who haunted the Macdonald lands. Travelers would often be found left mutilated by the creature. I can very much see that standing up to any dragon. Again, it's more a gameplay mechanic that doesn't really need to much thought behind it but if you wanted to, that would be the reason why.
I'd also like to clear up this misconception that faeries are all tranquil, pretty, or beautiful. Quite honestly if you've read anything about them. You'd never ever believe that their was any fairy that wasn't scary as shit. In legends and literature, Fairies, while they can be very beautiful are just as often evil if not more commonly, horrifying or incredibly disturbing.
Some like in Irish Folklore, specifically the Seelie Court are just indifferent, they like to play tricks and are sometimes even helpful. While the Unseelie court simply takes pleasure in causing as much pain and suffering in human's as they possibly can. And I don't mean just give you a cold. I mean torture you to death while you watch an ugly as sin creature that has no name (Traditionally because it's so horrifying and ugly that their are no words to describe how grotesque it is) slowly peel the flesh from your child's body.
Nintendo has so far, chosen the more modern Fairies are beautiful approach (Which by the way, originates from the Victorian Era) but that doesn't mean most faeries are like that.
For instance:
Anthropophagi
He's specially adapted to eating human flesh, since he doesn't have a head, and so has no eyes or ears to be swayed by cries from his victims. He has a gigantic mouth in the middle of his chest. It makes for easy passage to his stomach, I guess. He also lacks a nose, which is said to help him choke down human flesh without gagging.
Red Cap
Called a Red Cap because his hat is dyed red from human blood. He goes around decapitating people and their is said to be no escape from him.
I hope Nintendo includes the other side instead of just the cute ones.
