Why did Saul have authority to arrest Christians in Damascus?
Saul had authority because the Judaistic religious people had clout with Rome. Rome respected Judaism as an ancient religion and protected Temple offerings, so it permitted Jewish extradition of Jews for religious dealings, even from Damascus outside Israel, allowing Saul to arrest 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, and the Jews from one end of the Roman Empire to the other sent vast amounts of money to the Temple as offerings. Rome recognized and protected those gifts, so the Jewish authorities 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, which was outside Israel, and Rome permitted him to arrest Jews and literally drag them back to Jerusalem. Les also notes that this new thing was persecuted because it was something totally new and was treated like some kind of a sect.
And Saul, yet breathing out threatenings and slaughter against the disciples of the Lord, went unto the high priest,
More Questions
What was Saul doing to Christians in Israel before his conversion?
Before his conversion, Saul was trying to stamp out everyone in Israel who had proclaimed and believed Jesus was the Messiah.
Who were the disciples Saul was persecuting in Acts 9?
In Acts 9:1 the 'disciples of the Lord' that Saul was persecuting were ordinary Jewish believers in Jesus as the Christ, not the twelve apostles.