Ooh! The story about how it started's kinda more accepted than the urban legends I'm usually into, but it's a fun one!
So there was this village in Johto a long time ago, back before it and Kanto split into separate regions. No one was quite sure why or what they'd done, but for some reason the Nightmare Bringer would come to visit once a year on the full moon, tormenting all the people of the village through horrible dreams. Even worse, each time it happened one person wouldn't even wake up.
Then one year after the Nightmare Bringer's visit, this one girl from the village noticed that the wild Pokémon in the area didn't seem like they'd been affected at all, and she got an idea. If the whole village disguised themselves as Pokémon the night the Nightmare Bringer came back, it might think the village had been abandoned and leave without hurting anyone.
She told the village elders and they agreed that the idea was worth trying, and all that year the people of the village worked to make disguises for everyone before the Nightmare Bringer's next visit happened. And they finished just barely in time, two days before the full moon.
And then the full moon came, and everyone put on their disguises. They'd also agreed to stay up all night, just to make sure no one had bad dreams. And right at the darkest hour of the night, the Nightmare Bringer came through, looking for the people of the village, but all it saw was a bunch of Pokémon playing together.
And the sun rose with nobody falling asleep or having any nightmares at all! The girl who'd had the idea got praised as a hero, and everyone started work on fixing up the disguises they'd worn that night and making new ones for the kids that were gonna get too big for the ones they'd used before. And they kept dressing up like that year after year.
After a few years the Nightmare Bringer stopped coming, but by then the night had changed from a time of fear to a time of joy and partying, so nobody really wanted to give it up. And when some of those kids grew up and moved out of the village, they took the tradition with them, and eventually it spread all over the world!
And THAT'S why we all dress as Pokémon and stay up all night on the full moon in September. Or the first full moon if there's two that year.