Two reasons. First of all, Arab rulers often hired Turks for their armies, because Turks were strong and good soldiers. Turks fought for them, because well, it was a great way to earn money. However, Arab rulers grew overreliant on the Turkish soldiers in their armies, who grew into a military elite, which allowed the Turks to seize the real power in many Middle Eastern states because there was no military power which could oppose them anymore, which allowed them to use the Arab rulers as puppets or found their own dynasties. This is what happened with the Mamluks.

Second, as aforementioned Turks were good soldiers, as most peoples from the periphery and the steppes usually were. This allowed them to invade and conquer other realms. The Safavids managed to rally Turkic tribes called the Qizilbash around Ismail I based on his Shia heritage and conquered Persia. The Mughals similarly invaded India with an army of Afghans/Turks.

So, tl;dr, Turks were great warriors, so they either became the military elite and through that the political elite, puppeting rulers or founding their own dynasties, or they conquered their own realms.

You could see it as analogous to how widespread Norman-descended aristocracies became in places far removed from Normandy. A highly militarised ethnic group becomes disproportionately represented in a multicultural milieu during periods of endemic warfare.