“Listen - I know it's none of my business, but you should probably apologize to her.”
Your brother is having an argument with his wife. You think he is the one who is wrong in this case. You want to tell him that he should apologize to her, but you know that it's not considered proper to give couples advice on their arguments. Carefully, you give him this advice.
Listen - I know it's none of my business, but you should probably apologize to her.
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This is a conversational technique called a concession. Saying this shows that you understand problems that the listener might have with your advice or the idea you're trying to persuade them of.
"(somebody's) business" means decisions or details that that person is connected to and responsible for. The idea is that certain decisions are only one person's to make, and other people should not give advice or opinions on these topics. Stated positively, it's:
That's my business.
It's none of my business.
To "apologize" means to say that you're sorry.
"Apologize" uses "to" for the person who receives the apology, and "for" to indicate the reason for apologizing.
People often say "listen" before giving advice. It makes the advice sound honest and frank.