Why did Saul have authority to arrest Jews in Damascus?
Because the Judaistic religious authorities had clout with Rome, which respected Judaism as an ancient religion and protected the vast Temple offerings sent from across the empire. Rome therefore allowed them to demand extradition of Jews for their own religious dealings, permitting Saul to go to Damascus, arrest Jews, and drag them back to Jerusalem.
Les explains that the Judaistic religious people had clout with Rome. Rome respected Judaism because it was considered one of the ancient religions, while the Christian soul was treated as something totally new, like a sect. The Jews from one end of the Roman Empire to the other sent vast amounts of money to the Temple as offerings, and Rome recognized and protected it, so none of those gifts was lost. Their clout was enough that they could demand extradition of Jews they wanted to deal with in their own religion and bring them back to Jerusalem. That is why Saul had authority to go to Damascus, outside Israel, and Rome permitted him to arrest Jews and literally drag them back to Jerusalem.
And Saul, yet breathing out threatenings and slaughter against the disciples of the Lord, went unto the high priest,