Restaurants featuring this type of menu usually use such phrases as "down home", "country cooking" or other terms to convey to the patrons a quieter, less complicated lifestyle. Often the menu is presented to diners on numerous chalkboards situated around the restaurant within easy viewing of all of the tables
The "meat-and-three" (including the additional word "sides" will immediately clue the locals in to the fact that you aren't from around here) probably share their origin with the small town barbeque restaurants that are a mainstay of the small Southern town. Unlike the meat-and-three which is open all week long, the barbeque restaurants are usually open only Thursday through Saturday and serve enough food to feed a small army for the change you get out of your car's seat cushions.
In central South Carolina, the local meat-and-three is called Lizard's Thicket and features a green chameleon as its mascot. The origin of the name is not common knowledge, that I am aware of, but a t-shirt from said establishment makes for a dandy conversation starter in other parts of the nation.