It is generally the rule that the people who suffer conquest get influenced by the superiority of their conquerors and therefore see many of their individuals attracted to the culture and religion of the powers that defeated them.
But in this case, the victors are the ones who are influenced with the religion of the defeated. Berke actually made most of the famous golden horde convert to Islam officially and vowed to defeat Hulagu for the sack of Baghdad. What made him become such a devout Muslim with more allegiance to other Muslims than to his own Mongol people? as evident by reading the following declaration by Berke himself :
"He (Hulagu) has sacked all the cities of the Muslims, and has brought about the death of the Caliph. With the help of God I will call him to account for so much innocent blood."
Mongol leaders regularly converted to (or supported) the religions of the people that they subjugated. It was believed that this made them easier to govern. For this reason, Sorghaghtani Beki (mother of Möngke Khan, Hulagu Khan, Ariq Böke, and Kublai Khan), for example, encouraged the patronage of numerous religions. Similar policies were followed by all the early Mongol leaders. This has been attributed to numerous things; making governance easier, indifference, a method to avoid overreliance on a single religion etc. It is also likely related to the fact that Tengrinism hardly had the organizational apparatus to be forced upon all the conquered peoples. In any case, this sort of policy tended to lead to assimilation rather than domination.
The question asks what "convinced" Berke Khan to become a Muslim. It is unlikely that being "convinced" was involved. Most modern conversion scholars believe that claims that a religion's doctrine or beliefs were convincing to the convert are post-conversion narratives read back into their pre-conversion lives. But in any case, although I know about Mongol religious policy, I know little about Berke Khan so I am unable to answer that part of your question.
See:
Rossabi, Khubilai Khan: His Life and Times.
Forbes Manz, “The rule of the infidels: the Mongols and the Islamic world."
Jagchid and Hyer, Mongolia’s Culture and Society.
DeWeese, Islamization and Native Religion in the Golden Horde.
Well, I'd argue the point about conquest and the subjected people looking up to their invader's cultures as the norm. The Opium Wars were started because the Chinese did not want anything from the Brits while the Brits wanted tea and etc. Alexander was intoxicated by Persian Culture. No country converted to Shinto. There's also the relationship between religion and the divine right to rule. Many foreign rulers took to taking domestic titles, local political allies in marriage and the most successful were those that kept the freedom of religion.
You have to remember that Hordes were awarded to different sons of Genghis and all function as a different unit. It was Hulagu that took out the Persians while Batu and Berke were stationed North, looking for European expansion. The Golden Horde had already placed itself against the Ogedeids when Batu went against Guyuk. Mongke, a Toluid, rose to Khan with the support of Batu but died by disease battling in Southern China. This caused a civil war between Kublai, the second Toluid son, and the youngest, Ariq Boke, who, as tradition dictates, was stationed in the homeland. Hulagu, the third Toluid, supported Kublai and Berke supported Ariq Boke. Even then, Berke was set on not creating the precedent of Mongol infighting until the Ilkhanate cut off trade and stopped sharing the riches of Persia with its northern brethren, with the Golden Horde's reinforcements (albeit delayed by Berke) to Hulagu's invasion notwithstanding. It was not religion that drove Berke to war.
Anyway, Berke-Khan is to have run into a caravan from Bukhara (in Uzbekistan) and they convinced him to convert.
Here are some background information: