"To twist somebody's arm" is to force them to do something. You could force a person to do what you want them to do by twisting his arm and causing them pain until they do it. The phrase is used as a metaphor - usually what we mean is that you berate the person or make threats verbally in order to get them to do what you want. However, currently the phrase is almost only used sarcastically. It is usually said when somebody agrees to do something they already want to do. "A: Do you want some ice cream? B: Well, since you twisted my arm... (yes)." That is what is happening in your dialogue. It is such a common phrase that is understood to mean "Yes, but I will pretend that I don't really want to," that the agreement to A's request is entirely omitted. A would understand what B means, even if a non-native speaker does not.

To twist a person's arm means to persuade them to do something that they would have done willingly anyway. The dialogue could be revised to this to have a clearer meaning: A: Stay here tonight, please. It is your home afterall. B. OK , you don't have to twist my arm.