As with most questions related to religion, this is a difficult one to parse, due to the complicated relationship of religious texts and the historical record, but we’ll do what we can.
First, let’s strip the story of the Apostle Paul down to the most basic and widely-agreed-upon facts. As far as we can tell, a Jewish man named Saul of Tarsus emerged in the fledgling Christian Church in the years following the alleged death, resurrection, and ascension of Jesus of Nazareth. This man, who now professed Jesus as the long-awaited Jewish Messiah, travelled around the Roman Empire as an evangelist, and wrote a number of letters to the early Church that became Biblical canon and the bulk of the New Testament.
Let’s put aside the questions of how many of Paul’s letters actually belonged to him and how many were written by other Christians who slapped Paul’s name on their letter to lend it weight in the Church, since they are largely irrelevant to your question, and first ask this: was Saul of Tarsus a real person? The answer to this is an unqualified “almost certainly”. There is little reason to doubt that a man calling himself Saul in Hebrew and Paul in Greek was a prominent apostle, evangelist, and writer in the early Church. His miracles, conversation with Jesus, and events of his life are all up in the air for debate based on your personal interpretation of the Bible, but we can safely assume that yes, there was a man in the early Church named Saul of Tarsus.
Now, let’s get back to your original question - motives. Setting aside the Biblical explanation, as you requested, a surface-level examination of the cultural landscape seems to offer no reason for Saul to start playing for a different team, as it were. Saul was a self-professed Pharisee, a highly educated religious mover and shaker in Judea. The Pharisees, while holding little religious or political power, were nonetheless “in a sociological sense brokers of power between the aristocracy and the masses” (Evans 786). Even the anti-Pharisaical Biblical narrative implies that the Pharisees, while spiritually bankrupt and corrupt, were important figures throughout Judea.
The emerging followers of Jesus, on the other hand, held no such power and influence. Strange as it may seem in a predominantly Christianized world that we live in, the early Church was a much less widely accepted and marginalized group. They held no power, influenced no aristocrats, swayed no public opinion, and were largely seen as only a splinter group of Jews for several centuries - even among themselves. They believed that worship of Jesus was the next step of Judaism; they were simply in Judaism 2.0, rather than a new faith entirely. Of course, as the Church grew in prominence and attracted attention to itself, they would face opposition at various levels before being accepted and popularized by Emperors Constantine and Theodosius I, but all of this would happen centuries after Paul’s death.
So, we must return to the question of why? Well, a potential, if slightly cynical, hypothesis is that Saul was simply looking for the next horse to hitch his wagon to. We’re moving into areas of pure conjecture and hypothesis here, so sources will be a bit thin on the ground, but since our only insight into the mind of Paul is his own writings, we’re doing the best we can with the tools we have.
Saul, for all of his humility and claims that he was “the least of the apostles, unworthy to be called an apostle” (1 Cor. 15:9 ESV), was not above a bit of shameless self-promotion. After all, by his own hand, he was “of the tribe of Benjamin, a Hebrew of Hebrews; as to the law, a Pharisee; as to zeal, a persecutor of the church; as to righteousness under the law, blameless” (Phil. 3:5-6). Of course, Paul would be quick to point out that these things don’t really make him better than his audience, but it’s not hard to come away with a little sense of Paul flexing his Pharisaical privilege over these uneducated plebs. To use an analogy, Kanye West can rap about how his expensive purchases don’t make him happy, but there’s always an undercurrent of bragging that he has the money to do these things. What am I on about?
Back to Paul, it doesn’t seem to be out of the question that, if we look at the life of Paul with a cynical eye, he was attempting to join this smaller religious group with few leaders, and none as well educated as him, so that he could exert more influence over them than with his already-established Pharisees. The Pharisees had their doctrines, their leaders, their religious traditions, and reaching a place of prominence among the Pharisees would have been a much more difficult task for Paul than becoming the biggest fish in the small pond of the Christian Church. If Paul jumped ship and entered the Church, he would be able to wield a much greater amount of influence and call the shots, rather than just being another one of the Pharisees - a prominent group, to be sure, but a group in which Paul would have likely just been “another Pharisee”.
Of course, all of this is extremely difficult to prove. Our writings on Paul are scarce, to say the least, and all of them come with the historical caveat of being religiously-biased texts. Paul claimed that he gave up everything to follow Jesus because he loves God so much, but it comes down to his word against...no one’s. Alternatively, maybe Paul genuinely saw something of value in the emerging Church. Maybe Paul saw it as an alternative to the stale and religiously fractured Judaism of his day. Maybe Paul was a complete huckster and wanted to take advantage of people. Or maybe Paul really met Jesus and changed his life. But, in my personal opinion, divorced from my own religious beliefs about Jesus and Paul, this seems to be the most likely, secularist view.
If you’re interested in learning more about Paul, the early Church, and life in First Century Judea, I would recommend the following.
An Anomalous Jew, Michael F. Bird
Paul, a Biography, N.T. Wright
The Theology of Paul the Apostle, James Dunn
Paul, E.P. Sanders
*Christianity: The First Three Thousand Years”, Diamaid MacCulloch