DIAGNOSIS That baked-on grease and starch is polymerized—it's basically turned into a plastic-like coating from the high heat. Normal scrubbing after it cools won't touch it.
FIX You need to re-liquefy the gunk. The best method is a hot, soapy soak. 1) Fill your sink or a large tub with the hottest water you can get from the tap. 2) Add a good squirt of dish soap (Dawn is great for grease) and a half-cup of baking soda. 3) Submerge the entire basket and let it soak for at least 2 hours, or even overnight. 4) After soaking, use a non-scratch scrubby sponge or a soft nylon brush. The gunk should slide right off. For any stubborn spots, make a paste of baking soda and a little water and scrub gently.
IF THAT DIDN'T WORK For truly epic, neglected buildup, you can use the oven's self-clean function. Place the basket and tray on the oven rack, run the pyrolytic clean cycle, and let it cool completely. The ash will wipe off easily. Only do this with the metal mesh basket and tray, not any plastic accessories.
HEADS UP Prevent the problem: after each air fry use, while the basket is still warm (not hot), give it a quick rinse or wipe under hot water. A little prevention saves a huge cleaning headache later.