This is an entirely complex question to answer. Probably the best way to start is to explain why it was not even uncommon for slave dynasties to arise in medieval Islamic societies.

First and foremost, I'd just like to point out that slavery in medieval Islam is not necessarily as lenient as what you wrote makes it out to be. I'm no expert on the subject, but a slave remained their owner's property and even with certain rulers on how to treat slaves, it's still a terrible life. That vast majority of slaves would work on fields, industries and other such economic activities their owners would have them do, or fall into household work.

Only a specific subclass of slaves had a very particular, relatively beneficial at times, relation with their owners. These are the people who rose to rulership. Those Mamluks or Ghulams (which also means slaves) were the military elite of medieval Islamic societies.

The institution of slave-soldiers as the military elite came during the early Abbasid period when it became necessary for the Caliph to have an army of professional soldiers loyal only to him, rather than dependent on the aristocracy. How this was accomplished was by purchasing slaves from terrtiories outside of Islam (the overwhelming majority at that time came from the steppes, from pagan Turkish tribes). These slaves were brought into the royal household, were taught, were indoctrinated, were converted and trained to become soldiers. They formed the professional core of the army and some of these soldiers, by their loyalty and dedicated service, would also be given high command positions and eventually become amirs (rulers, governors).

The importance of these slave-soldiers as the Caliph's household army, and eventually the household troops of most Islamic rulers, was due to the regional ambitions of governors and the nobility, which had ties to their region and could muster support, which could prove a serious problem for the Caliph. These slaves were cut off from their families and were brought to a region where they had no ties. They were quite often disliked by the locals. Their existence and prosperity depended entirely on their master's ability to keep his position, and possibly to expand it. So, as time went on, with proven loyalty and service,some were granted manumission (freed), high positions, iqta's (a tax farming title) and other benefits.

As the Abbasid Caliphate splintered (partially due to Ghulam commanders playing kingmaker with the Caliphal seat), regional rulers took over and also built up their own household armies of Ghulams. The same process would happen in other regions, with Ghulams being given large administrative and military responsabilities.

The first major slave dynasty was that of Ghaznavids, which arose out of the Samanid Amirate, centered originally in the Afghan region of Ghazna. The Seljuks would develop a similar household army, and as the empire fractured, more Ghulam amirs would take over regions and carve out their kingdoms.

Now, to the Mamluk sultanate. As the Ayyubid dynasty was a confederation of members of Saladin's family, it led to a large amount of power struggles. As Ayyubid power collapsed in Egypt, the large and powerful Mamluk factions took advantage and grabbed power. In a sense, it was not any more uncommon than what had happened elsewhere in the past centuries. Like in all slave dynasties across the Islamic world, the Mamluk dynastyies of Egypt and Syria had a big problem with dynastic stability.

So if I can summarize, how these slaves managed to acquire such power is due to the dependence of Muslim rulers on these slaves as their military elite and most trusted members of their household. They gained power from their service to their masters and when the ruler's power eroded or collapsed, these people, being the most powerful members of the military and the aristocracy, they were in prime position to take over.

Sources :

Cambridge history of Iran

The military technology of classical Islam, David Nicolle

The Delhi Sultanate : A political and military history, Peter Jackson

Plenty of other documentation that I can't think of specifically. Anything by David Ayalon is excellent.