‘And the chief captain came and said to him, “Tell me, are you a Roman?” And he said, “Yes.” And the chief captain answered, “With a great sum obtained I this citizenship.” And Paul said, “But I am a Roman born.” '
So the chief captain came and officially put to Paul the question as to whether he was a Roman. To answer untruthfully to that question would be a serious offence. But when Paul replied ‘yes' he had cause to be afraid. Roman citizens had to be treated with care. Questioningly he said, ‘Such a citizenship cost me a great deal of money'. Paul replied, ‘But I was born a Roman citizen.' That made clear that he came from a distinguished family, for he was born and bred with citizenship rights.
We know from Acts 23:26 that the name of the chief officer was Claudius Lycias. He had probably therefore bought his freedom when citizenships were being sold off by the favourites of Claudius. Prior to that time citizenships had been more exclusive and given for especially meritorious service. Thus he knew that Paul's ancestor must have been at the very least a very important official who was seen as loyal to the emperor.
It should be noted that for someone to claim to be a Roman citizen when they were not was a capital crime, and made them subject to summary execution, and as his citizenship could be proved or otherwise from citizenship records it would be foolish for a non-Roman citizen to make such a claim (each citizen was certificated on birth, a certificate which would be kept in the family records, but it may even be that they carried with them a certificate of citizenship. We actually know little about the details).