The rise of the Turks in the Islamic world stems from two sources: inside the Abbasid Caliphate and Outside. Internally, the Caliph and other governors started importing large number of young Turks as slave soldiers given that as the Turks lived outside the Dar- Al Islam (The Turks didn't truly incorporate themselves into the Muslim world until the 10th century), they had no ethical loyalties that characterized the Arabs or Persians and were loyal to the Caliph or governor alone (similar to how the Roman Emperors used Germanic warriors as bodyguards.)

Eventually, these slave soldiers became more and more powerful as their numbers and position increased and in many places, decided to take power for themselves and overthrow the old ruler. The first of these dynasties were the Ghazvanids in what is today southern Afghanistan, who rebelled and overthrew the native (Persian) Samanids, who had de facto control over the region. This also happen in Egypt in 1250 (though a lot of those slave soldiers or Mamluks are of Circassian or northern Caucasus stock).

The second and ultimately more influential way the Turks incorporated themselves into the Muslim world was by invading it. The Ghazvanids mentioned earlier over the late 10th- early 11th century had to contend with the Seljuk Turks who at the time still lived on the steps. Long story short, the Seljuks defeated the Ghazvanids in 1040 and started taking control over the Muslim, reaching their height in 1071 after defeating the Byzantines (Romans) at Manzikert (I go into detail about how the Turks also incorporated themselves into the power struggle following Manzikert here: http://www.reddit.com/r/AskHistorians/comments/2wnwbz/if_the_battle_of_manzikert_was_not_a_bloodbath_as/cosm847)

The Great Seljuk Empire, as this entity has come to be known, was inherently unstable which combined with targeted assassination by the Assassins broke the Empire apart and various Turks started claiming individual cities in the Near and Middle East (a huge reason the First Crusade was so successful).

In short, the Turks were able to assert themselves as the rulers of the Muslim world by their military might either from going from Slave Warriors to rulers via coup or by military conquest and migration ( Anatolia is very similar to the Turkic homeland in the steppes and so one of the reasons Turkey came to be prominently Turkic)

Well, much of those are the legacy the Seljuk Turks. Their empire did not last long, but it left a legacy of Turkish governments in the Middle East from Erzerum to Fars, with some examples being the Sultanate of Rum (http://en.wikipedia.org/wiki/Sultanate_of_Rum) and the Khwarazmians (http://en.wikipedia.org/wiki/Khwarazmian_dynasty).

Aside from that, they were given great power in the Abbasid Golden Age as slave soldiers (Mamluks) and were instrumental in enforcing the power of the Caliph. With the power of Mamluk troops, the Fatimid Caliphate (the only major Shia caliphate of the Middle Ages) kickstarted the current Sunni-Shia conflict that exists today.