Peaceful world in which kids used to play and live in peace. Many years ago, Xehanort was born here. When he was a teen, he believed it was too small. He was greeted by his Heartless to time travel to find 12 versions of himself. He came back to the present. He escaped to Land of Departure through unknown reason. The night Sora, Kairi and Riku were planning on leaving, the islands got consumed by darkness as heartless started coming in. The world is made back to normal at the end of the first game when Kairi goes back. We have a small glimpse at Destiny Islands again at the end of Kingdom Hearts 2.
All the characters would always come to their own little island near the main island to play, free of adults. When Sora and Riku were kids, they spotted a shooting star one night. They eventually find out that Kairi was the one arriving to the islands for the first time. Kairi was originally a resident in Radiant Garden, until it was overrun by Heartless, and she made her way between worlds, eventually ending up in Destiny Islands. Kairi claimed she had no memory anything that happened before that.
The three came to be close friends and made their own adventures in the island they played in. But then Riku dreamed of leaving the island, strongly believing in the possibility to travel to other worlds. Sora, Riku and Kairi then built a raft and gathered supplies for their trip. Suddenly, the night before they were to set off, the Heartless discovered and invaded their home. Sora, believing it was a storm coming, went to check on the raft. When he arrived on the island, he noticed that Riku and Kairi were ahead of him, and he went around the island to look for them. He finally found Riku, who was welcoming the catastrophe that was devastating their island, and let himself be consumed by darkness. Sora tried to reach out to him, but Riku disappeared. Suddenly, the Keyblade appeared in his hand. It was revealed that Riku was the original who was chosen to wield the Keyblade. But when he allowed darkness to consume his heart, it was passed on to Sora. Sora then went on to look for Kairi while fighting off the Heartless. He then went in their secret hideout, where he found Kairi, not feeling herself. A huge gust of wind blew from the mysterious door in the secret hideout. Kairi was blown towards Sora, but disappeared in a split second she reached out to hug him. Sora was blown outside, and fought off a giant Heartless, and after the battle, he was blown off Destiny Islands before it was finally consumed by the Heartless completely.
After Sora, Donald and Goofy traveled to different worlds and fought the Heartless and eventually defeating Ansem and sealing Kingdom Hearts, the worlds, including Destiny Islands, was restored. Kairi returned to the islands, but separated from Sora, and promised he would come back to her once he found Riku again.