The Fat Boys came out in 84 right around the same time as Run DMC, Whodini & UTFO, all managed by Russell Simmons. They were on the first Fresh Fest tour, which was the first Hip Hop tour.
Those groups were the second wave of Hip Hop, the first wave being Sugarhill Gang, Grandmaster Flash, Kurtis Blow, Fearless Four, Crash Crew, Cold Crush, Spoonie G and all of the groups that released records from 79-83.
When Sucker MCs dropped producers stopped using live musicians and started using drum machines and synthesizers. Rap took on a different sound. The early rap records were produced like R&B songs, but circa 1984 rap music began taking on its own identity. By 1984 rap didn't sound like R&B anymore.
Basically Russell Simmons saw what Sylvia Robinson was doing wrong. Sylvia and Sugarhill records were trying to appeal to the R&B audience. They had ballads on their albums and they dressed like punk rockers. When Run DMC, Whodini, The Fat Boys and UTFO came out, they dressed more Hip Hop, like B-Boys.
But alas, whoever started managing The Fat Boys turned them into a novelty group. They had skills like Run DMC and Whodini, but they started making gimmick records like Wipe Out, Louie Louie and The Twist. They also made a few movies. Krush Groove was dope but they made another movie called Disorderlies which was kinda corny.
They were dope for the first two albums but they got progressively worse and by their 3rd and 4th albums they were just a novelty act.
Having said that, this is my favorite song by them
Here's a few more
Edit. BTW, that Fat Boys album in the O/P belongs in The Vault