As far as I am concerned, there are three theories about the conversion of Turkish people into Islam.
One of them states that their 'Tengriism' was already close to the doctrines of Islam, so they easily accepted the religion as their own when they met Arabs.
Another theory defends that Turks were made muslims by the might of the sword. Especially Curcan and Talkan massacres are said to cause the masses to convert, as a means of saving their lives.
The last theory I know is a bit more simple. It asserts that the Turkic Beyliks (chiefdoms) of the time wanted to have better political and diplomatic relations with the Arabs, therefore, they lead their people into Islam.
Which one could be correct? Or is there a fourth theory?
The Turks began to convert to Islam en masse in the 800s and 900s. The main cause behind this was the fact that the Abbasid caliphs in Baghdad imported Turkic warriors from Central Asia to serve in their courts. First as bodyguards and later as generals and advisors. As individual Turks emigrated to Baghdad to serve in the Abbasid (and its successor states') military, entire families and tribes followed.
Their conversion to Islam was a result of the immense social, economic, and political benefits of being a Muslim in that time and place. Just like the Persians and others who converted out of real secular benefits that were attached with being Muslim, the Turks converted slowly and managed to rise to the top of Middle Eastern society, best seen through the Ghaznavid and Seljuk Sultanates.
The beyliks of Anatolia developed much later, in the 1200s, a few hundred years after the conversion of the Turks to Islam.
Sources: A History of Medieval Islam by JJ Saunders The Venture of Islam by Marshall Hodgson