The movie was shot pretty much entirely in a studio, using an on-set airplane cross-section and airport terminal set. In fact, there's a specialty movie studio just for this purpose: Air Hollywood. Founded by a film producer who experienced first-hand the problems of filming aviation scenes in a real airport, Air Hollywood provided both the cabin and flight deck sets for Airplane! (Source)
Throw in some occasional green screen and a not particularly realistic looking model jet for the exterior shots, add the accompanying propeller plane sound effects for good comedic measure, and that's all there is to it.
From the score to the camera work and lighting, to the casting, down to the performances themselves, ZAZ sought to emulate the hyper-serious pomposity of popular '70s disaster films. But when coupled with the filmmakers' special brand of absurdism and self-referential humor, the result is comedy gold.