In American slang, a 'sloppy joe' is a loose-fitting sweater/sweatshirt, a kind of hamburger or a scruffy person. However, given that you ask: 'What is Sloppy Joes'?...in the singular and with capital letters, I'd guess it might be a caf�, club or whatever.
I know Sloppy Joe's was a fashionable restaurant/nightclub in Havana that was frequented by the US 'well to do' before Castro took over. There is one now in Key West, Florida, but I don't know if this was started up by the person who owned the Cuba one after the revolution.