The brilliance was using actors, who primarily played serious roles, especially Leslie Nielsen.
Now you can’t watch him in his earlier serious roles without laughing out loud.
Leslie Nielsen was known as a "serious" dramatic actor up until Airplane! which made him "the Olivier of spoofs" according to film critic Roger Ebert, and leading to further success in the genre with The Naked Gun and Scary Movie film series, and a leading role in a Mel Brooks comic horror, “Dracula: Dead and Loving It” – IMO not Brooks’ best work but still funny.
Robert Stack, Lloyd Bridges and Peter Graves also showed great comedic talent in Airplane!