Why did Saul ask the high priest for letters to the synagogues in Damascus?
Because he wanted to stamp out the belief that Jesus was the Messiah by persecution and imprisonment, and he was not satisfied with only the small towns of Israel; he sought authorization to go into Gentile territory, to the synagogues in Damascus, to find any followers of "this way" and bring them bound to Jerusalem.
Les explains that Saul of Tarsus, a good Jew, saw the believing Jews who followed the heresy that Jesus of Nazareth was the Messiah as a threat to Israel. He was already trying to stamp it out by persecution, imprisonment, and death. Not satisfied with working the small towns of Israel, he wanted to go into Gentile territory and go to the synagogues. He asked the high priest for letters to Damascus so that if he found any of this way—that Jesus was the Christ—whether men or women, he might bring them bound back to Jerusalem.
And Saul, yet breathing out threatenings and slaughter against the disciples of the Lord, went unto the high priest,
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Who did Saul get letters from to go after believers in Damascus?
Saul went to the high priest and requested from him letters addressed to the synagogues in Damascus, giving him authority to act there.
Why did Saul go to Damascus to arrest Jews instead of the Syrian population?
Saul's target in Damascus was not the Syrian (Gentile) population; he wanted only the Jews who had embraced faith in the Messiah.