“How rude of me! Karen, this is my friend Jeff.”
You're at a party. You were talking to one person (Karen), but then another friend of yours (Jeff) comes over to talk to you. You've been speaking with Jeff for a few minutes, and you suddenly notice that Karen is uncomfortable because she doesn't know him. Realizing your error, you say this.
How rude of me! Karen, this is my friend Jeff.
You use this phrase to sort of pretend to be shocked by something. When you're really shocked, you say things like "Oh my God!" and so on. But when you want to show other people that you're shocked or moved in a mild way, you can describe the situation like this:
How nice of you!
How stupid of me!
How horrible of him!
The basic meaning of this phrase is "What (someone) did is so (adjective)". So "How rude of me!" means "What I did was so rude!"
This is a polite way to introduce two people to each other. First you say the first person's name. You start with the person you were just talking to. In this example, you were talking to Jeff but you said "How rude of me!" to Karen, so you start:
Then you say "this is" and then describe the relationship:
this is my sister's husband
this is my colleague
this is the president of the University