It's a great question. I can speak a bit to the Golden Horde and the Ilkhanate, although I have to admit being rusty on this; I haven't read up on Mongol conversion literature since university.

It's a good question because both the "how" and the "why" parts are interesting. Let's start with the "why."

You can take this in at least two directions. The first is, why did these rulers convert how and when they did, what were their motivations for doing so. The second, I think more difficult question is why was their religion seemingly more susceptible for them to be converted than for them to convert others.

There's a great essay looking at this question from a comparative perspective by Anatoly Khazanov called Muhammad and Jenghiz Khan Compared: The Religious Factor in World Empire Building. It looks at what at first glance looks like two very similar sets of circumstances: rapid conquest of a vast empire by a charismatic tribal leader bearing a unique religious/political claim to rule. But where as Muhammad and his successors Islamicize and Arabize the conquered territories, the Mongols do not succeed in a similar cultural and religious conversion of their empire. Instead within a few generations they had largely adopted the religions and cultures of the people they conquered.

Khazanov looks at a lot of interesting reasons for why that is (it's well worth a read), but the conclusion I would emphasize is simply that the Yasa of Genghis Khan, the closest thing they had to a religious scripture, just doesn't make much sense as a religion for conquered people. The text itself was literally kept secret and its scope was limited. As Khazanov writes:

In all probability, the main theme of the Yasa was the necessity to maintain the unity of the royal clan and of the Mongol empire under the sway of a single ruler (Ayalon 1971). The Yasa also emphasized military discipline. That was all, or almost all.

So the first part of an answer to why the Mongols converted is that their own lack of a coherent religious doctrine, let alone a proselytizing one, left them in a sense "vulnerable" to religious conversion.

The second answer I think has more to do with the political circumstances of the territories that they had conquered. The general consensus is that the Mongols where they converted to Islam did so for political reasons. What were those political reasons?

Well, for one thing there was sheer demography. Though the areas conquered by the Ilkhanate and the Golden Horde were religiously diverse, Islam predominated. My understanding is that this was less true in the Russian steppes conquered by the Golden Horde, but it would have been true of Turkic steppe peoples that were most readily suited to joining the Mongol polity.

Just as important I think is that Islam by this point included a pre-packaged set of institutions designed precisely to cater to nomadic tribal peoples ruling over settled lands that they had conquered. The notion of nomads ruling settled peoples had been the norm within Islamdom for hundreds of years at this point and in response had developed what Marshall Hodgson called the "Amir-Ayan system."

The Amir-Ayan system was a model of governing in which the nomadic rulers would hold military authority, but notables from outside the tribe would hold religious, legal, and other governing authority. This was an incredibly effective system for ensuring that tribal military chaos, which saw a sometimes rapid rise and fall of tribal dynasties (e.g. the Samanids, the Ghaznavids, the Buyids, the Ghurids, etc.), did not result in societal chaos. In fact, the centuries following the disintegration of centralized Abbasid rule were quite prosperous in Greater Persia. The system worked fine so long as the tribes that were succeeding one another were all Muslims.

The Mongols obviously were not. And the results were deeply chaotic. For one thing the conquests themselves were hugely destructive, with whole cities wiped off the map. But even in administration the pre-Islamic Mongols were a bit of a disaster. They don't seem to have understood the complexity of agrarian life in Persia, which among other things relied on constant government maintenance of a system of underground aqueducts known as qanats.

Adopting Islam and adapting to a system that was primed and ready to go just made a lot of sense. I don't think it's a coincidence that the conversion of Ghazan to Islam and his empowerment of a Persian vizier/prime minister, Rashid ad-Din, also led to incredibly successful administrative reforms and had a stabilizing effect on Mongol rule over Persia.

Now, ostensibly at least the Mongols already at the time of the conquests of the western areas of greater Persia had dropped their traditional beliefs. Hulagu, who sacked Baghdad, was a Nestorian Christian (which he inherited from his mother) and converted on his deathbed to Buddhism. My understanding however is that their religious convictions at this point were largely nominal.

That was also apparently true of the initial conversions to Islam. Berke Khan of the Golden Horde for instance, apparently converted to Islam publicly while privately maintaining the Yasa.

This I think is a good moment to transition to the how of the conversion, which I think is interesting. A lot of focus is rightly given on the political nature of the conversions of the Khans, but there's reason to think that a conversion process was already happening. There's debate as to how that was, with some scholars emphasizing the role of itinerant sufis in propagating the faith. The Mongols also practiced mass conversion. When Ghazan converted, whole tribes converted as well. I don't think we should necessarily discount this as entirely cynical. In tribal societies, mass conversions are pretty common, historically speaking. The other vector which I alluded to earlier is the interaction between the Mongols and the already Muslim Turkic steppe peoples who were often integrated into their own armies, particularly in the Golden Horde and Ilkhanate.

Sourcewise, I've mentioned a few books and authors who it would be worth reading, but it would also be worth checking out Devin DeWeese's Islamization and Native Religion in the Golden Horde: Baba Tükles and Conversion to Islam in Historical and Epic Tradition. It's been a while since I read it, but I recall it being good. Also, just for a short introduction to the region and its governance, David Morgan's Medieval Persia is good.