Ekhard Schnabel asks this question in Paul the Missionary (44, cf. Early Christian Mission, 2:927-928). There are rally two questions here. First, what was the theological motive for Paul’s persecution? Second, what drove him to pursue Jesus’ followers to Damascus?
Some scholars have argued the Jewish Christians were admitting Gentiles without circumcision. This seems unlikely, since there is no reference at all to Gentile mission by the Jerusalem Church until Acts 10. God-fearers were accepted into the synagogue without circumcision, so it is unlikely this would be a problem for Paul, if it had occurred. Similarly, some argue Gentile believers were not concerned with food traditions. This too is unlikely for the same reasons as the first, there is no evidence of Gentile converts in the pre-Pauline period. These two issue are a problem only when a significant number of Gentiles were saved, and especially Gentiles who were not God-Fearers before accepting Jesus as Savior.
A more likely motivation is the possible political / social problems caused by the preaching of a crucified messiah / savior. How would this play before the Gentiles, especially the Romans? Could this be an accusation against Rome, and a possible rally-point for anti-Roman activity? The problem here once again is the lack of evidence for preaching anything to Gentile / Roman audiences. The early apostolic mission was confined to the temple area and the city of Jerusalem in general. Remember that the factors which will eventually result in the Jewish War are already in the air some thirty years earlier. Paul may have been concerned for sparking a revolution by teaching that Jesus is a resurrected King who will return and establish a kingdom.
It is probably best to see Saul opposing the Apostolic teaching as heretical. That Jesus was the Messiah was absurd, since he was crucified, “hung on a tree,” and therefore a curse, not salvation. In addition, Schnabel points out that any theology which saw Jesus as Savior is not compatible with the view that salvation comes through faith expressed in obedience to Torah. A simple example from the gospels will illustrate this point. When the rich you man asks Jesus what he must do to inherit eternal life, he understands this in terms of obedience to the foundation of the Law (ie., he keeps all the commandments). This is not to say Judaism was a “works for salvation” religion, but that one was right with God because God has given Torah and individuals come to God through the perfection of the Torah.
These early followers of Jesus claim that there is no other name by which a person can be saved (Acts 4:12). Stephen’s speech in Acts 7 concludes with a contrast between the Torah and Jesus. Saul’s motivation is to correct this false teaching within Judaism, using the synagogue punishment system itself. He likely sees himself as a reformer, working for the high priest, with the goal of dealing sharply with the followers of a condemned Rabbi.