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.