Why did Saul go to Damascus to arrest believers?
Saul went to Damascus because the High Priest and Israel's rulers had extradition authority with the Roman government over certain cities; they sought to bring back renegade Jews who had followed Jesus of Nazareth, so they could deal with them, imprisoning and putting them to death.
Les explains that Judaism had clout even with the Romans, and the High Priest and rulers of Israel had enough influence that certain cities of the Middle Eastern Roman Empire would honor their demand for extradition, much like modern extradition between America and Canada. Saul and the religious leaders therefore set in motion a plan to go to Damascus and bring back the renegade Jews who had followed Jesus of Nazareth. As Paul later rehearses in chapter 26, they threw them into prison and put them to death. This occurred before Saul of Tarsus met the Lord on the road to Damascus.