If you look up the following scriptures on biblehub.com you can see the versions that use this phrase (some King James and I believe there is another one) Acts 2:37, 5:33 & 7:54 (this is all I have found so far).
The meaning however is not of an emotional one but of truth.
We see in scriptures like Jeremiah 18:12, Exodus 4:21, Isaiah 6:10 (and many others) God speaks of a 'hardened heart' -therefore, when the Apostles were ministering the truth (or anyone; because it's the Holy Spirit who brings the truth to each person regardless of the era) 'cut to the heart' is the description that tells us 'in that moment God was able to reach them with the truth' and from there we see how the crowd or persons acted.
In chapter 2 of Acts they asked Peter what should we do and he said 'repent'. Then in chapter 5 & 7 the men were furious (basically that they were called out AND they KNEW it was the truth) so their actions that were moved by their evil hearts followed.
