English Standard Version (ESV)
Have you ever been in a position where you really didn't want to hear the truth? Maybe you knew you had done something wrong, or maybe you were coasting along in a good place, and you just didn't want to hear that everything wasn't as good as it looked like. I've had some moments where my car was broken and I knew it but I kept driving anyway -- just turn the radio up and pretend that everything is ok! The repair would be costly, and I wanted to pretend I didn't need it.
Here, the leaders of the Jews had convinced the people that Jesus was a fraud, and they didn't want to hear Peter's message that he was the Messiah, but they needed to. The only way to find a solution was to accept the reality of their brokenness -- it's a necessary step for them to understand that the only way forward was to recognize how wrong they were and that Christ was truly who he claimed to be.
We, too, are in the same position. We are sinners, lost without hope, and then Christ reaches down and redeems us from sin. He saves us by his sacrifice on the cross, and because of his selfless love, we have hope. The only way we can have this hope, though, is to recognize that there is something fundamentally wrong -- we are sinners in need of a Savior, and we cannot save ourselves. Only Christ can do this, and once we accept this truth we can receive his grace and be reconciled to God.