I can't speak to exactly how it made its way to Hungary, but as a gamer and Christian who's been both since the 1980s I can explain the general history of religious backlash against fantasy role-playing games.
The Church was initially quite uncomfortable with acting and theater back in the early ADs, and with fiction writing in general as well. It was hard for them to distinguish fiction from lying (see the movie Galaxy Quest for more). Keep in mind that in the Christian faith, thinking about committing a sin is sinful in and of itself.
But I say to you that everyone who looks at a woman with lustful intent has already committed adultery with her in his heart.
And he said, “What comes out of a person is what defiles him. For from within, out of the heart of man, come evil thoughts, sexual immorality, theft, murder, adultery, coveting, wickedness, deceit, sensuality, envy, slander, pride, foolishness. All these evil things come from within, and they defile a person.”
As a result, "pretending to be a murderer" is sometimes seen as a slippery slope - are you not contemplating murder, thinking about it and how you'd do it and why... Of course magic and worshiping spirits and whatnot are all mentioned in the Bible and considered both quite real and sinful in traditional Christianity, so pretending to do those has the same close equivalence to actual sinning in the mind. And worshipping gods other than God is one of those "Christianity 101" no-nos. Over time, these new activities were further understood and allowed for by more nuanced theology, to where no mainstream church has an issue with acting or fiction writing per se today, although they do believe they can be used to promote sin and sinful thinking.
Role-playing games emerged from wargaming. No one cared.
There was a huge scare about Satanic ritual abuse in the 1980s. Various people, mostly of dubious authenticity, testified to being victims or perpetrators of various "old school" Satanic/witchcraft kinds of activities. Not all of this was false, of course, as there's freaks in every time period, and the publicity caused some amount of copycatting (little different from the spread of TM/Yoga in the 1960s, for example). This received an epic amount of media attention and created a backlash against anything that could be seen as occult-related in nature. D&D was very popular at the time; it had become a large fad (complete with TV cartoon) - the D&D Basic set (Red Box) sold millions of copies and was in every chain bookstore. TSR sold $20 million worth of product in 1982 alone! Since D&D talks about pagan gods and sorcery and whatnot, and it was a high profile target that would add to media attention, it was immediately picked up on as a target by certain parties like Patricia Pulling and William Schnoeleben who crusaded against the game publicly. This led to a series of controversies, culminating in a 60 Minutes feature in which Gary Gygax appeared and, unfortunately, did a spectacularly hapless job of defending the hobby on the show. The Tom Hanks movie Mazes and Monsters, which popularized the suicide of James Egbert III, a disturbed boy who was a gamer, along with other media exposure like the 60 Minutes interview caused some questions about the game in non-religious circles as well, which caused religious efforts to ban the game to get much less resistance than would usually be experienced.
Even those Christians who did not buy into the claims of real Satanism and spells in the game could often be swayed by the "sinning in the heart" argument, as it requires time and effort to understand both gaming and theology enough to effectively discern the truth from the hearsay and it's often easier to just tell your kids "don't play that." This isn't necessarily ill-intentioned, as a parent myself I understand that it's like other common practices of judging appropriate media for your children based on reviews and other parents' comments; you can't watch every single thing yourself first and have to go with "what you've heard" out of efficiency. As D&D is a group activity, it only takes 1 in 10 parents, school officials, priests, etc. against it to ban it in many venues. Even if the official in question doesn't think it is evil, if some parents complain they have some responsibility to act, even in Scripture it admonishes "Abstain from all appearance of evil." (1 Thess 5:22).
The primary push was against D&D, but was generally broadened to "role-playing games" since D&D was the largest by far (most people had never heard of others) and since many of those others were similar and didn't necessarily change the argument - MAGUS, for example, would be open to the exact same criticisms as D&D.
Reactions from within the gaming community included the renaming of demons and devils to baatezu and tanar'ri in the Second Edition of Advanced Dungeons & Dragons. They tried to keep such language less religiously charged until the late 1990s. Counterattacks were mounted by gaming groups - Michael Stackpole wrote pieces like this one rebutting and discrediting Pat Pulling. The Committee for the Advancement of Roleplaying Games was founded in 1987 to combat people trying to discredit RPGs.
As the satanic panic died down, the reaction against D&D did as well, though like any urban myth it still lingers. I was publications director of the Christian Gamers Guild for a time, they have a FAQ written about gaming directed at addressing questions from Christians about things they've heard are "evil" about gaming. It is still an issue sometimes in particularly conservative, rural communities. Waves like this tend to take time to spread, so the panic and then recovery felt in other parts of the world were delayed (much like the advent and departure of disco :-). You still see flareups of anti-RPG propaganda today, though much like burning Harry Potter books, it's limited to an extremely small fringe of people largely agitating about anything popular to get on the news to spread their faith.
However, the damage still lingers. Here's a 2004 poll on a Catholic forum in which 11% (of an admittedly small sample) felt that online RPGs were not OK for Catholics. That hits that one-in-ten level that's likely to create some complaint that leaders often humor by disallowing things. There has never been an official statement on RPGs from Catholic leadership, though in 2000 they did issue a statement rebutting concerns about Pokemon cards, which were the next popular inheritor of the "is it sin?!?" bellwether. Today, the discussion has moved on to video games, and whether our violent video games promote sinful behavior.
See also the same question on Christianity.SE: Source of Christian attitudes towards fantasy and role playing games