"The fact that God raised him from the dead, never to decay, is stated in these words: 'I will give you the holy and sure blessings promised to David.' So it is stated elsewhere: 'You will not let your Holy One see decay.' For when David had served God's purpose in his own generation, he fell asleep; he was buried with his fathers and his body decayed. But the one whom God raised from the dead did not see decay." NIV translation
Paul now uses the words of Isaiah and David to show the people the promise that God fulfilled through Jesus Christ. He is showing them that death was conquered through Jesus and his resurrection.
"Therefore, my brothers, I want you to know that through Jesus the forgiveness of sins is proclaimed to you. Through him everyone who believes is justified from everything you could not be justified from by the law of Moses. Take care that what the prophets have said does not happen to you: 'Look, you scoffers, wonder and perish, for I am going to do something in your days that you would never believe, even if someone told you.'" NIV translation
Paul ends his history lesson with the reason for the history which is salvation through Jesus Christ. He explains that God knew there would be "scoffers" who would not believe and be saved and Paul begs them not to be one of them. The quote is from Habakkuk the prophet and comes after Habakkuk has asked God why he allows the injustice in the world.
God's reply was that he would bring judgement that the prophet would not even believe. God was referring to the coming invasion by the Babylonians.
"As Paul and Barnabas were leaving the synagogue, the people invited them to speak further about these things on the next Sabbath. When the congregation was dismissed, many of the Jews and devout converts to Judaism followed Paul and Barnabas, who talked with them and urged them to continue in the grace of God." NIV translation
By simply presenting the good news (gospel) to the people and trusting God to work, Paul set a very good example that is missed today. Paul didn't have a fancy planned out sermon with bullet points and the like; he simply told people about Jesus. Too many "preachers" today are trusting in their words, wisdom, and works instead of allowing God to work. We see that, in this passage, God worked on the people's hearts and caused them to want to know more about him and his Son; it wasn't anything about Paul or his wisdom and strength.