I'm not the best person to ask, but I'll give it a shot.
We've only found them in Wisconsin, but I think it's a Midwest thing. IOE, they're typically well themed restaurant where locals go to meetup. The first one we went to felt like we had traveled back in time to the 30s to 40s complete with a few older guys acting like mobsters hitting on younger girls at the bar.

That's the coolest one we've been to though. Most of them just have a nice atmosphere. They typically have above average food & drinks, where multiple courses are common. I think there has been at least one free course with the meal at all the supper clubs we've been to. It may be as simple as a palate cleanser or significantly more. I think they've all offered a bread service, which isn't common at most restaurants we frequent these days. All that said, our experience with them is very limited. I hope someone who grew up in the midwest will chime in with a more accurate description.