To twist someone’s arm literally means to take a person’s arm and turn it around, which could be really painful if you take it exactly word for word. If your arm has been twisted it means that someone has done a great job of convincing you to do something you might not have wanted to to do.

And if you manage to twist someone else’s arm it means that you’re great at convincing them, and they’ve finally agreed to do something after you’ve been begging them.

Tom: Jake you should really come to the party tonight!

Jake: You know I can’t, I have to hit the books (study).

Tom: C’mon, you have to come! It’s going to be so much fun and there are going to be lots of girls there. Please come?

Jake: Pretty girls? Oh all right, you’ve twisted my arm, I’ll come!