Acts 2:37—“When the people heard this, they were cut to the heart and said to Peter and the other apostles, “Brothers, what shall we do?”
In Acts 1:4-5 After his resurrection Jesus told his disciples: “On one occasion, while he was eating with them, he gave them this command: “Do not leave Jerusalem, but wait for the gift my Father promised, which you have heard me speak about. For John baptized with water, but in a few days you will be baptized with the Holy Spirit.”.
After this occurred, what we know as the Day of Pentecost, the disciples began to speak in unknown tongues, yet everyone understood what they were saying. Some, even though it was only about 9:00 am, thought they were drunk.
In Acts 2:14 Peter explains to the crowd what’s going on. Quoting Joel 2:28-32 he reproves the crowd bringing clarity and truth to the false accusations: “Let me explain this to you; listen carefully..this is what was spoke by the prophet Joel I will pour out my Spirit on all…Your sons and daughters will prophesy, your young men will see visions, your old men will dream dreams. Even on my servants, both men and women, I will pour out my Spirit in those days, and they will prophesy…”
He goes on to explain how "Jesus of Nazareth was a man accredited by God to you by miracles, wonders and signs, which God did among you through him," and that he "was handed over"....put to death on the cross, "But God raised him from the dead..". Peter furthermore briefly recounted Israel's history and how Jesus surely was the Son of God and Messiah they had hoped for.
Upon hearing this truth, then, as referenced in the comments above, (The Book of Acts, chapter 2) "When the people heard this, they were CUT TO THE HEART and said to Peter and the other apostles, “Brothers, what shall we do?”
Peter replied, “Repent and be baptized, every one of you, in the name of Jesus Christ for the forgiveness of your sins. And you will receive the gift of the Holy Spirit. The promise is for you and your children and for all who are far off—for all whom the Lord our God will call.”
I believe what "cut to the heart" means then is, what they knew, thought to be true, better yet, what was held dear or close to the heart, was not true. Or at least is no longer true because of the new information given to them: the truth of salvation through Jesus Christ and the forgiveness of sins.