Mamluks were not slaves. They were once slave-soldiers who completed their training and were freed, becoming 'professional' soldiers and quite possibly a local ruler named by the Sultan or Atabeg or whatever title the master had.

A slave-soldier (Ghulam) was trained for years and owed loyalty only to his master, something they are well-known for. Once completing his training, he was freed (as a Mamluk, 'owned'). Since they were incredibly loyal, it was natural for rulers who depended on them to assign them to positions such as emir. Once the rulers became weak, such as the case of the Ayyubids, Ghurids, Abbasids or Seljuqids, some Mamluks were in a prime position to take over the Sultanate or the parts of the Empires/Kingdoms they ruled for the Emperor/Sultan, due to a combination of significant political position, resources, lack of central authority or power of said central authority, local support and/or support from military leaders, as Mamluks were not only potential local rulers, they generally rose to the higher echelons of the military.

In this sense, they're not much different than local dynasties rising to become kings or emperors, just a very different background.

Sources :

• Military Technology of Classical Islam, Nicolle

• The Crusades : The Authoritative History of the War for the Holy Land, Asbridge

• The Cambridge History of Iran

• The Delhi Sultanate : A Political and Military History, Jackson