Why were the magistrates of Philippi alarmed after beating and jailing Paul and Silas?

Answer: They were Roman citizens

Paul said they had been beaten publicly without trial though they were Roman citizens, so the magistrates came to appease them.

“But Paul said to the officers, “They beat us publicly without a trial and threw us into prison, even though we are Roman citizens. And now do they want to send us away secretly? Absolutely not! Let them come themselves and escort us out!””— Acts 16:37 (BSB)