What does Acts 13:22 mean?

Paul is giving the history of God's saving work in the history of Israel. Of all the "saviors" in Israel's history, King David is surpassed only by Moses. From the time that David was a boy and killed Goliath (1 Samuel 17) throughout his reign as king, he consistently saved the nation from their enemies.

There is understandable confusion about how David can be a "man after God's own heart" despite his infidelity with Bathsheba and murder of her husband Uriah. These are serious sins and God dealt with them (2 Samuel 11:1—12:23). The moral aspect of David the man, while important, was not God's priority for the king of Israel. His priority was that the king would obey Him and honor Him as the head of the nation. In this, David excelled:

1. David never showed any indication that he was even tempted to worship an idol, despite allowing his wife to have a household idol (1 Samuel 19:13). David only ever worshiped God.
2. David valued and defended God's honor, even above his own. We see this in his reaction to Goliath (1 Samuel 17:45–47) and in his humility when his son Absalom usurped the throne (2 Samuel 15:24–26).
3. David took very seriously God's charge to Israel that if they obeyed and honored Him, they would have success in war, and if they dishonored Him, they would lose. David saw his battles as an extension of God's mandate to Joshua's generation to conquer the Promised Land. It wasn't until he was king that Jerusalem was taken from the Jebusites (2 Samuel 5:6–9).

Yes, David the man was imperfect. David the father was too lenient. But David the God-follower was sincere and steadfast. And David the king was the only king of a united Israel who honored God's covenant with His people. More to Paul's point, God used David to save His people.

There is also some question about the quote as it seems to be a mash-up of several Old Testament passages. This is apparently a quote from the Septuagint, but from Paul's memory, as the words are not exact.