How did Peter explain the healing of the lame man in Acts 3?

Peter explained the lame man's healing as occurring through the power of Jesus Christ of Nazareth, the one the nation had crucified. He pressed Israel to repent of rejecting their Messiah and other national sins, noting that if the nation repented, God would then institute the prophesied tribulation.

Les explains that Peter and John had dealt with the lame man, and the nation of Israel was upset about how the healing happened. Peter and John tried to explain that it was through the power of Jesus Christ of Nazareth whom they had crucified. In verses 19–21, Peter said that if the nation would repent—primarily of the awful sin of rejecting their Messiah, as well as their other national sins—God would have immediately instituted the tribulation. The tribulation had to come because it was in prophecy.

Why did Peter say the lame man was healed through Jesus Christ 'whom you crucified'?

Peter proclaimed the healing came through Jesus Christ of Nazareth, whom Israel had crucified, to confront the nation with its awful sin of rejecting their Messiah. He used the miracle as a call to national repentance, showing their guilt and God's continued offer of mercy if they turned back to Him.