"Then Paul stood up, and motioning with his hand said, 'Men of Israel, and you who fear God, listen: 'The God of this people Israel chose our fathers, and exalted the people when they dwelt as strangers in the land of Egypt, and with an uplifted arm He brought them out of it. Now for a time of about forty years He put up with their ways in the wilderness. And when He had destroyed seven nations in the land of Canaan, He distributed their land to them by allotment. After that He gave them judges for about four hundred and fifty years, until Samuel the prophet. And afterward they asked for a king; so God gave them Saul the son of Kish, a man of the tribe of Benjamin, for forty years. And when He had removed him, He raised up for them David as king, to whom also He gave testimony and said, 'I have found David the son of Jesse, a man after My own heart, who will do all My will.' From this man's seed, according to the promise, God raised up for Israel a Savior--Jesus--after John had first preached, before His coming, the baptism of repentance to all the people of Israel. And as John was finishing his course, he said, 'Who do you think I am? I am not He. But behold, there comes One after me, the sandals of whose feet I am not worthy to loose'" (Acts 13:16-25).
Paul was more than willing to speak a "word of exhortation" to this group of people who had gathered at the synagogue. It appears that there were both Jews and Gentiles (proselytes) gathered by the way in which Paul addresses them. He begins with a history lesson (which is a great way to make important connections between the present and the past) going all the way back to the final days of Jacob where the Israelites had dwelt in Egypt, initially as strangers and later as slaves. God blessed them greatly in that land (despite their suffering) and brought them out of the captivity with a mighty hand. He tolerated their stubbornness in the wilderness for forty years and carried them along, so to speak, like a mother does her child. They were forced to wander for that duration as a punishment for their weak faith. He then enabled the next generation to conquer seven nations in the land of Canaan and distributed land to each of the Israelite tribes. As needed, judges were raised up by God to lead the people (particularly in battle). After several hundred years, the prophet Samuel came along. At their request, Samuel, against his desire, gave them a king--Saul. Saul was later replaced by David, a man after God's own heart (i.e., one who would be characterized by humility, not rebelliousness). From David's lineage would come the Savior--Jesus the Christ (and this was predicted in several places; cf. II Sam. 7:12,13; Isa. 11:1; Psa. 132:11). John the baptizer announced the coming of the Messiah and prepared the way for Him. John admitted that he was not the Anointed One and that he was not even worthy enough to unloose His sandals (which was the task of the lowest household slave)! John was looked up to by the people as a great prophet. His testimony about Jesus was significant and persuasive. Why is Paul going this route to introduce them to Jesus of Nazareth? It likely was the most expedient way to get their attention and assure them that this message was not something entirely new but that it was intimately connected with the old message they knew. In fact, it was an extension and fulfillment of it.
As a side note, the reference in 13:20 (in the NKJV) to the period of judges lasting 450 years cannot be reconciled with I Kings 6:1. I believe the ESV has a better rendering here, and it reads - "All this took about 450 years. And after that He gave them judges." The 450 year period is describing the interval from the choice of "our fathers" (often dated from the birth of Isaac) to the distribution of the land in Joshua's day. Such is consistent with other chronological data contained elsewhere in Scripture.