They were not exempt from state laws. They just essentially ran their own municipality as opposed to being within one run by a nearby city or county. This also meant that they had to operate the government organizations of a municipality for the Reedy Creek Improvement District like police and fire, including taxes, etc. in addition to operating everything for the park itself. However, since the land owned by Disney was the only thing in the district, that wasn’t really an issue.
The only reason you don’t really see this elsewhere is that amusement parks usually aren’t built on 27,000 acres of empty swampland with nothing else nearby. Amusement parks are almost always built near existing cities and municipalities already governing the areas. Disney World was built in the middle of a swamp with nothing nearby originally for it to affect.
The Florida State Legislature in 1967 created the Reedy Creek Improvement District after Walt petitioned for more autonomy so they could build EPCOT, which was originally going to be a fully planned “community of tomorrow”. After Walt’s death the company decided not to get into city-building however, so EPCOT became more focused on the idea rather than actually building a city.
It’s been operating semi-independently ever since, paying taxes, maintaining roads, police, fire, etc. just fine while allowing Disney to expand the park as needed. The only reason it became an “issue” was DeSantis not liking Disney’s politics and deciding to go after them for daring to speak out against his politics, which the US Supreme Court has continuously said over the years is clearly protected by the First Amendment. Notably his first attempt to “punish” Disney was going to saddle nearby counties with millions of dollars in debt and requirements to manage police, fire, etc. which they simply wouldn’t be able to do. No thought went into it whatsoever, it was simply a State Governor wanting to punish a company for publicly being against his politics.