Does it mean 'heart was attacked with anxiety/anxiousness'? Therefore in the example it would mean people's heart would be anxious?
It's a sudden heart attack by something unpleasant and unexpected, let's say a bombshell.
I don't know anything about Traditional Chinese Medicine or 'excessive internal heat'.
Reading this explanation, I think you could, at least in some contexts, simply translate '急火攻心' as 'stress'.
Basic explanation refers to the mental illness caused by a person's lack of proper venting due to some uncomfortable things in life