This not a blanket statement. Otherwise, people can do whatever they want to others and wash off any responsibility. It varies according to specific situations.

In the example you give, yes, you would be the one causing hurt to others.

However, if you had arranged to meet up with a friend then you had a personal emergency. You apologise but 5 months later, your friend is still giving you a hard time. Are you then responsible for their feelings?

If you unintentionally cause someone pain or simply make a mistake (e.g. you forget an appointment with your friend) - you apologise but they won't let it drop. Are you forever responsible for their feelings?