-자 is the 'let's' ending, -자고 quotes it, and 그러다 stands in for 말하다. Every line here reports somebody else's proposal: 디너 크루즈 가자고 그러지 않으셨어요? (didn't you say let's go on the dinner cruise?), 사귀자고 그러면? (what if he asks you out?). The textbook form -자고 하다 means the same; 그러다 is the colloquial substitute you will actually hear.