Prelude to Panic
The groundwork to the 80s panic was probably laid by the 1979 disappearance of Dallas Egbert. A child prodigy with ties to the RPG subculture, unable to fit in, attempted suicide, attempted to run away from home and, a year later, shot himself. He was thought to have played D&D, and that part of the story was what the media focussed on between September 2nd and 14th.
Role-playing games were blamed and the connection was probably fixed in some peoples' minds after a Tom Hanks movie pushing the same narrative entered theaters in December 1982.
Six months prior to the premiere of the film, Irving Pulling shot himself, his mother found Role-playing books among his belongings - she had not been previously aware of his hobby.
In 1984, William Dear published his book The Dungeon Master, in which he recounted his involvement and search for Dallas Egbert back in 1979.
National Panic
The next year, the panic started to truly form. To quote Wikipedia,
In 1985, Patricia Pulling joined forces with psychiatrist Thomas Radecki, director of the National Coalition on Television Violence, to create B.A.D.D. (Bothered About Dungeons and Dragons). Pulling and B.A.D.D. saw role-playing games generally and Dungeons & Dragons specifically as Satanic cult recruitment tools, inducing youth to suicide, murder, and Satanic ritual abuse. Other alleged recruitment tools included heavy metal music, educators, child care centers, and television. This information was shared at policing and public awareness seminars on crime and the occult, sometimes by active police officers. None of these allegations held up in analysis or in court. In fact, analysis of youth suicide over the period in question found that players of role-playing games actually had a much lower rate of suicide than the average.
(I have found 1982, 1983, and 1985 as dates for the founding of B.A.D.D., but Radecki seems to have involved himself and lent credence to the idea in 1985; he has since been stripped of his medical license and is currently serving a prison sentence for trading opioids for sex)
Patricia Pulling, having lost her son to suicide, latched onto his RPG hobby as an explanation. Unfortunately, this proved a profitable narrative for the media, just as much as it had been in 1979.
To quote Gygax on 60 minutes in 1985,
There's no link, except perhaps in the mind of those people who are looking desperately for any other cause than their own failures as a parent, for their child's death.
Here's a retrospective article on the topic, and another on the Dallas Egbert case.