It is the easiest, to view the context. People tend to postpone unpleasant tasks, therefore may need frequent reminders. Example:

Wie muss soll ich dir noch sagen, dass du dein Zimmer aufräumen sollst? (How often do I have to repeat, that you shall clean up yoour room?)

So the task must be quite a pleasant one, if no reminder is necessary at all. A typical use is:

Nimm doch das letzte Stück Kuchen!

So the response

Dass lasse ich mir nicht zweimal sagen!

indicates something like it is a pleasure to comply, typically even exceeding that, i. e. that you wanted to eat the cake anyway, but were too shy to ask for it. So I agree with Devon, that your example is not the typical one, which is a response to a request.

It is the easiest, to view the context. People tend to postpone unpleasant tasks, therefore may need frequent reminders. Example:

Wie oft muss ich dir noch sagen, dass du dein Zimmer aufräumen sollst? (How often do I have to repeat, that you shall clean up yoour room?)

So the task must be quite a pleasant one, if no reminder is necessary at all. A typical use is:

Nimm doch das letzte Stück Kuchen!

So the response

Dass lasse ich mir nicht zweimal sagen!

indicates something like it is a pleasure to comply, typically even exceeding that, i. e. that you wanted to eat the cake anyway, but were too shy to ask for it. So I agree with Devon, that your example is not the typical one, which is a response to a request.