This is a competition BBQ term.
A turn in box is the box with the meat turned in (hence the name) for judging.
The boxes are typically 9×9 white Styrofoam containers like the ones for your leftovers to take home from a restaurant. Each box has a number on top representing the BBQ team.
For a Kansas City Barbeque Society (KCBS) sanctioned competition there are a minimum of four meats turned in: chicken, pork ribs, pork butt/shoulder, and brisket. Garnish (typically lettuce and parsley) may be used to improve the appearance of the turn in box.
Here are examples of turn in boxes…