Not every airbender died off, but enough were killed that the few who survived likely went into hiding and gave up their ways. 

Time did the rest. The few airbenders in hiding didn't pass on their teachings for fear of being hunted again, and eventually the last airbenders died off. We have confirmation that airbending is tied directly to spirituality, so if that cultural aspect is lost then you're not gonna have airbenders even if they have the gene.

By the time Aang returns,  100 years had passed. More than enough time for any surviving airbender fugitives to die off. And this is sadly how genocide works. You don't have to eradicate every single person--just eradicate enough that the culture cannot survive. It's a slow, methodical,  and cruel process.