Something that's airborne is flying through the air. A soaring seagull is airborne, and so is a rock launched from a slingshot.
If you've ever tried parachuting from an airplane, you've been airborne, sailing through the sky, and when the plane itself takes off from the runway, it becomes airborne. On a windy day at the beach, people's umbrellas and towels sometimes become airborne. Bourne here is the past tense of bear, or "carry," so when something is airborne, it's carried by the air.