The Cotton Club was originally founded as the Club Deluxe in 1920, by Jack Johnson, the first African American world heavyweight boxing champion. He'd previously opened a racially inclusive restaurant/club with his second wife, who was white, in 1912. Following that, he was prosecuted for, basically, having sexual relationships with white women, and he and his wife went abroad for the rest of the decade. He returned in 1920 and presumably made arrangements for opening the Deluxe shortly before turning himself in and serving his year in Sing Sing. He would sell up in 1923 to Owney Madden, a big-time gangster and bootlegger, who renamed it "the Cotton Club".

(This is according to most of the sources I have found. According to Vaudeville Old & New, however, Johnson turned it into the DeLuxe Cabaret in 1924 and shortly thereafter it was taken over by Madden. I can't find primary sources to corroborate either.)

Either way, Johnson was a huge public figure and any club he was associated with would attract attention, so it's not surprising that it drew people, who would then continue attending once it was under new management.

The original conception of the Deluxe was simply a high-end, glitzy nightclub, but Madden had it renovated under the direction of the set designer for the Ziegfeld Follies into the racist plantation-themed nightmare that was the Cotton Club: the stage made to look like the porch of a mansion, the backdrop with slave quarters. Madden poured money and illegal liquor into it, which continued to contribute to its attractions. He also had the clout to encourage famous performers to work there, another form of glamour, and had composers like Harold Arlen writing songs that would become classics for the stage shows. As a result of the liquor and excellent performances, wealthy and well-known celebrities were known as regulars, some of whom were drawing a crowd who didn't just want to be entertained and drunk, but to rub elbows with American aristocracy.

And ... you can't underestimate how much white audiences liked the plantation theme. For many white Americans, there was a lot of nostalgia for the supposed simpler and happier time of the antebellum south, and the nineteenth century in general. (See this answer of mine on the "gay '90s".) Romantic clothing, chivalric behavior, blacks "knowing their place", etc. Add that in with the famously beautiful, light-skinned dancers and showgirls, and it's not surprising that it was successful.