"To leave the airport" means to exit or go outside the area where airplanes land and take off, usually to continue your journey elsewhere. Full definition
Before leaving the airport, ask the airline if they will deliver the bag without charge when it is found. (transportation.gov)
Make sure you track your puppy's flight prior to leaving the airport in case of a delay / change. (premierpups.com)
Generally, each of the individual clubs feature dishes by local celebrity chefs unique to each city, which is a great way to sample regional fare without leaving the airport! (travelcodex.com)