Years ago, I participated in the filming of an episode of Unsolved's sister show, Rescue 911. (I'm one of the skinny 6th graders hula-hooping in the gym while a kid gets his tongue stuck in a canteen.) They filmed at our real school, with the actual participants, but what confused me at the time was that virtually none of the details of the story as filmed were accurate. (We were all outdoors for Field Day. The kid had been cracking wise with his friends about the forbidden topic of French kissing!--so you can see perhaps how the real incident fell together.
Presaging the era of 24/7 reality tv, the major American networks in the late '80s/early '90s were riding high on the popularity of docudramas. This wasn't a new format by any means, but it was enjoying a renaissance. 306
It could match the look and tension of hour-long dramas without the added cost of actors, sets, and a full writing room. The stories that aired weren't exactly true, but they were true enough for entertainment purposes.
So why the stark disclaimer? True, as commenters have pointed out, "Unsolved Mysteries" wasn't afraid to juice their ratings with some truly ridiculous UFO/bigfoot/ghost segments, (They were certainly my favorites, as a kid!) but on a base level even the ho-hum episodes played fast and loose with the incidents of real people's lives. At the time, network news was still loosely governed by the Fairness Doctrine--technically rescinded in 1987, but no one wanted to tempt an outrage that could push Congress to bring it back. Docudramas might not have presented many "controversial issues of public importance," but they wouldn't have stood up to the standards of network news at the time, and anyone aggrieved by their portrayal could have instigated a (costly) Federal Communications Commission suit.
Basically, it was a shrinkwrap-sticker attempt to head off lawsuits, under the guidelines that American network tv operated under at the time, like the "do not aim at face" warning on a Nerf gun. Probably not necessary, probably not enforceable, but a zero-cost attempt to cover one's rear end. 862