It is nearly always used with negative as "I don't give a damn", meaning 'I don't care', 'I'm not interested'. Your sentence means "I think you have mistaken me for someone who cares"; a sarcastic way of saying that you're not interested about what someone is saying. You can use the saying in some circumstances in the positive; "In fact, I do give a damn about the environment..."

I give a damn = I care I don't give a damn = I don't care I believe you are mistaking me for someone who gives a damn (=cares). This is a common expression in English.