Opening up with "I'm sorry" sets a tone for the conversation. If my last interaction with someone was a little tense, I need to _reset_ the emotional balance before moving forward.

In this hypothetical, I don't believe I did anything explicitly wrong. But I also didn't handle the situation well. Maybe the "sorry" is "sorry for not addressing your concern". Or maybe it's just "sorry that 'mansplaining' is a situation you encounter frequently". It doesn't have to be BLAME. Just empathy.