Translation:They asked him to take them to the airport.
"To the airport" means you are moving there, and that's "na letiště". If you are already there, it's "na letišti". So if you are taking someone TO the airport, it's "na letiště", if you are taking them at the airport (like taking them to your car or whatnot), it's "na letišti".
And you don't use "do letiště/v letišti", it's just a wrong preposition. It's always "na letiště/na letišti". It's in similar vein as saying "I'm in airport" in English, just doesn't work for arbitrary reasons.