How did the Turkification of Byzantine empire take place?

I’m wondering how the old Greeko-roman culture of Byzantine, perhaps the most sophisticated in the middle ages, just collapsed and replaced by language and culture of tribal immigrant Turks who hardly had any written tradition?

For the specifics of what happened in Anatolia, see When and how did modern Turkish become the majority in Anatolia? (where I’m about to post an answer).

For the general principle: people don’t adopt a different language and culture because they’re choosing between Cyril of Alexandria and Avicenna, or between Michael Psellos and Ibn Tufail. People adopt a different language and culture because it will bring them direct benefit in their daily life, in the regime they happen to find themselves. And we’re talking a lot of peasants in Anatolia, who had never heard of Psellos or Cyril.