The only time I’ve ever heard this phrase is in connection with describing a type of male hairstyle called a “mullet.” In the mullet hairstyle, the hair is cut short in front (so that it looks appropriate for a serious, businessman’s haircut) but long in the back (like the person is a party-er, not a serious business-person). So, the hairstyle gets jokingly described as “business in the front, party in the back.” The mullet hairstyle is considered an object of ridicule.

There are a large number of American crime movies which show illegal gambling and drinking in the back room of a business. So, a store in a crime movie may have "business in the front" and "a party in the back." The clearest examples are crime movies set in the 1920s during the prohibition period (when alcohol was prohibited). At that time, illegal drinking clubs and gambling clubs were widely tolerated in the US.