Ooh!! A theory topic! He he! ~Ahem~ Well...
Our world and the world of the Mario games are two separate dimentions connected at certain points by Warp Pipes, one of these places being Brooklyn, NY. Mario and Luigi stumbled across this on a plumbing outing. They originally lived in Brooklyn, but after being summoned back to the MK by Toad and Peach for the events of SMB3, they have since set up shop and stayed in the Mushroom Kingdom.
The Koopas do NOT come from another dimention. They are simply another race that occupies the same world as the Mushroom People, along with Yoshies, Frogs, Moles, and the occational Human. The Koopa Kingdom is big, but far less appealing than the MK, so not only is Bowser trying to take over the MK, but many Koopas have reformed and live in the MK (ala Monstro Town, Koopa Village, etc).
Bowser is a Koopa. The Koopa race includes most of the 'bad guys' from Mario games, like Koopa Troopas, Goombas, Buzzy Beetles, etc. Specifically, you could say Bowser and his kids are a special Koopa breed resembling a cross betweem dragons and turtles. They are Royal Koopas!
I love theory topics! Check out some of my others further down the page. And, for the record, I agree that the mystery of Mario is half the fun. Everyone can have their own ideas, and it's okay!
I'm done...for now!
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I''ve decided to add a signature. Now...what should I type? Hmm...