What does Acts 13:37 mean?

Paul is speaking to a synagogue of Jews, along with Gentiles who fear God, near the center of modern-day Asia Minor. He finishes his argument that Jesus of Nazareth is the Savior God promised would come from the line of David and that His death and resurrection were prophesied in Jewish Scripture—particularly by David. Like Peter (Acts 2:27–31), Paul uses David's words in Psalm 16:10: "For you will not abandon my soul to Sheol, or let your holy one see corruption."

This promise can't apply to David because he died, was buried in a tomb in Jerusalem that could be visited at that time, and never left that tomb. Conversely, three days after the Jewish ruling council orchestrated the crucifixion of Jesus and two friends buried Him, God raised Him from the dead. His resurrection was witnessed by His disciples, including five hundred men at one time (Luke 24:13–35; John 20:11–29; 1 Corinthians 15:6; see Acts 13:30–31). His body was not only not corrupted, but it was also restored to the point that some of His disciples struggled to immediately recognize Him (John 20:15).

So far, Paul has had the relatively easy task of reviewing God's saving work in Israel's history and promise of an ultimate Savior (Acts 13:16–25) and convincing his audience that Jesus of Nazareth is that Savior (Acts 13:26–37). Now, he sets out to properly define "salvation." To this point, it has included rescuing the Israelites from slavery in Egypt, giving them a homeland, and defeating their enemies. God's promise that His Savior will sit on David's throne and bring peace and prosperity to an independent Israel is yet to come. First, however, God offers a different kind of salvation: forgiveness from sins and freedom from the Mosaic law (Acts 13:38–39).