"God has raised this Jesus to life, and we are all witnesses of the fact." NIV translation
Peter's speaking to this crowd of Jews is based on Jesus' empty tomb. Prior to this passage, Peter explained that King David had a tomb where you could go and still find his decaying body. Then he explained that Jesus was not in the tomb where he was placed upon his death. Here Peter is telling them that all of them are witnesses to the fact that the tomb was empty. This fact was undeniable because all the people could go to the tomb and see that there was no body there. Logically, there can only be a few explanations for the empty tomb: 1. Jesus was not dead when he was placed in the tomb and he pushed the boulder out of the way and escaped. 2. Someone came and moved the boulder and stole his body. 3. He was resurrected and God removed him from the tomb. The first two explanations do not make sense because the tomb was guarded by Roman soldiers. They would have been killed for allowing the body to be stolen and, in fact, they were scared when it was discovered that the tomb was empty.
"Exalted to the right hand of God, he has received from the Father the promised Holy Spirit and has poured out what you now see and hear. For David did not ascend to heaven, and yet he said, 'The Lord said to my Lord: "Sit at my right hand until I make your enemies a footstool for your feet." NIV translation
Peter continues to contrast Jesus Christ and David in the first part of these verses. He explains that Jesus was not only raised from the dead (which is something that happened to others such as Lazarus) but he, unlike King David, ascended into heaven. He then goes on to explain that the miracle that the crowd is seeing is caused by Jesus pouring out the Spirit which was foretold in David's quote from the Psalm.
He then gives further proof by quoting Psalm 110 and basically says that, since David is still in the tomb, he could not have been talking about himself in this quote. Therefore, it must be Jesus that King David was referring to.
""Therefore, let all Israel be assured of this: God has made this Jesus, whom you crucified, both Lord and Christ."" NIV translation
Now, Peter is getting personal. He is basically telling the Jews that, because of their own scriptures, they can be sure that Jesus is the promised messiah. That is basically what the term "Christ" means. You see, the Jews believed the Torah (Old Testament) and the fact that there was a coming messiah. They just did not believe that it was Jesus and that was the problem. Even today, many Jews wait for the promised messiah.
"When the people heard this, they were cut to the heart and said to Peter and the other apostles, 'Brothers, what shall we do?'" NIV translation
We notice that, when the people were presented with the simple facts, they asked what shall we do. Many times we, as Christians, get caught up in thinking that we have to know so much of the Bible in order to reach someone. Here, we see that Peter basically shared that Jesus was crucified, buried, and resurrected. That was it and then God convicted the people, not Peter or any of the apostles. There was no debate or fancy plan to share the gospel. It was a simple presentation of the facts and trusting in God to do the rest. We should all remember that and do likewise.