“I know it's none of my business, but...”

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. You carefully say:

Listen - I know it's none of my business, but you should probably apologize to her.

Listen -

People often say "listen" before giving advice. It makes the advice sound honest and frank.

I know (a fact), but (clause)

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.

it's none of my business

"(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.

Stated negatively:

It's none of my business.

apologize to (someone) for (something)

"Apologize" uses "to" for the person who receives the apology, and "for" to indicate the reason for apologizing.