This is a very important topic, which many points reaches even farther than apologizing. The essence of what you're writing, as I see it, is communicating on a emotional level. If you talk to someone on any personal matter, and only focuses on some external thing (say, something that's happened, been said, and so forth), you don't connect with that person on a emotional level. If on the other hand you actually see the person, and recognizes the persons feelings, that person will more likely feel that it's need for support have been met.
These thing reminds me of non-violent communication, that I'm a big fan of.
By the way, I took the liberty to link to my own post, which touches up on some of the same things: https://steemit.com/psychology/@protoken/when-someone-is-late-for-your-appointment-you-get-either-relieved-or-annoyed-how-come
I guess they both touch on the topic of getting to know oneself.
You got it.