I have studied a lot of Turkish history, end been very interested in Turkish genetics.

One of several reasons is that a large amount of Nomadic Turkic people settled in Anatolia. They were called Yoruk Turks. According to Ottoman archives, in 1520 160,000 out of 860,000 households were considered as nomadic Turks. Instead of simply ruling over the pensula, the region received a wave of migration from Central-Asia.

Source (short read)

The region also had gone through a process of Hellenization (Greekification). Anatolian people adopted the greek language and their religion, and later Christianity. I don't know the details of the process, but a believed reason of this process is because the Greeks had a sphere of influence over Anatolia with the conquest of Alexander The Great. The people may have prefered using the 'lingua franca' of that time.

Back to Turkification with some side-info. The Ottoman culture already was derived from the Seljuk culture, which was a combination of Turkic and Persian culture. Turkic people already adopted a culture of a region they conquered.

Another reason a lot of Anatolians, and Balkan people, Turkified is because of the blood tax. Non-muslim citizens had to donate a child to the Ottoman state, whom were converted to Islam and fought for the Sultan as Devshirme.

During the late 19th century, the Ottoman Empire endorsed the Turkish culture and language. This lead later to the emergence of the Young Turks, who would later declare the Turkish Republic and endorse, even force, the Turkish nationality.

Now with all this in mind, I explained how Anatolia Turkified. The why can simply be explained by Islam. Oghuz Turks were already Islamized before they arrived in Anatolia. The rule of thumb for Nomadic people to convert to the region they conquered, was not compatible with the goals of Islam at that time. Churches were converted to Mosques and people were endorsed to convert to Islam. Together with Islam, came the Turkic culture.

Edit: Minor correction