The plot for Sonic 3, Japanese version, is kicked up another notch from those of the previous games, and the island of choice this time is Angel Island (aka "The Floating Island" elsewhere in the world, at the time). Angel Island cannot normally be found on the sea, it has a remarkable ability to float high above in the air, due to a mysterious power which would be discovered in Sonic & Knuckles. Nevertheless, this mysterious island of legend is abruptly taken out by a certain gargantuan chunk of metal falling from space, and mountains are crushed and forests are burned as a result of this unfortunate collision. Together, they fall back down to the ocean where Angel Island rests on the surface. The name of the space junk that caused this catastrophe? None other than the Death Egg, huge spherical satellite of doom, created by Dr. Eggman (Robotnik, in the west) himself. Sonic managed to knock it out of commission at the end of Sonic 2, but having had its fall broken by the island, it may now be within the possibilities of repair and re-launch, as it sits comfortably on a lake within the island.
Though he doesn't fully realise what happened, the events do not go unnoticed by one Miles "Tails" Prower, who witnessed a large tsunami wave occur, caused by the giant splash made by the island when it crashed into the sea. Days later, Tails picks up strong emerald readings on his radar, from out in the ocean. Since he and Sonic still hold all seven Chaos Emeralds collected from their last adventure, this was no doubt puzzling, so he decides to consult with his friend. Sonic was enjoying some relaxation on the beach when he finds a mysterious ring with strange carvings written on it, washed up from the sea. The ring has no particular significance to anything, other than reminding our hero of a story he had once heard...
According to the legend, there once lived an ancient race of individuals, thousands of years ago, dwelling on a now lost island. They used a mysterious "mighty stone" to help advance their society, and made regular worship to it, until one day, when a faction of greedy elders of the tribe intended to steal its power all for themselves. What exactly happened then is unclear, but it's believed that these actions caused the entire population to be wiped out, and what was left of their architecture and culture remained on the island, which magically rose high up into the skies, where it would remain hidden forever from future generations. This inspired Sonic for another adventure, and after hearing of Tails' breaking news, the duo geared up their Tornado bi-plane and took off towards the source of the energy readings, Chaos Emeralds in hand.
How little they know of what lies ahead, however, because Dr. Eggman has not only already begun to repair his Death Egg on the island, he's also found a new ally in the shape of Knuckles the Echidna, last remaining resident of the island, and guardian of the secret it holds within. Due to his lonely life and lack of social interaction, it's all too easy for the doctor to make him believe anything he tells him. Anticipating the arrival of Sonic and Tails, gullible Knuckles is made to believe that they're the bad guys, come to steal the source of the Chaos energy all for themselves. With immense strength and power on his side, plus knowledge of every hidden path and secret on Angel Island, Sonic will surely meet his match upon arrival!
As with Sonic 1 and 2, here in the Western world, we get a version of this prelude that's largely the same idea, but with the details lopped off. Unlike the Japanese version, this account does seem to imply, however, that Angel Island has its own set of Chaos Emeralds which keep it in the air, presumably separate from the original set being carried by Sonic. According to this version of the story, it's the island's emeralds that Robotnik wishes to steal for himself, in order to get the Death Egg back up and running. Of course it's up for debate, but to avoid confusion, it's commonly believed that there is in fact only one set of seven emeralds in the Sonic series, and that the earliest records of them suggest that their place of origin is within Angel Island, alongside the "mighty stone" of legend. The reason Knuckles steals Sonic's emeralds at the start of Sonic 3 is either as a kind of favour for Eggman, or to restore them to their original location. In Sonic Adventure, we find out what really happened to Knuckles' lost race, and what it was that destroyed them.
As a footnote, check out Studio Ghibli's Laputa: Castle in the Sky, if you ever get the chance. Apart from being a brilliant animation anyway, you'll spot some similarity between its castle and Angel Island. They both float high above the clouds as the stuff of legends and are both held up by a single mysterious jewel, without which, they fall apart.