싫어하다 is disliking something, and -게 with 이렇게 or 그렇게 in front of it measures the degree — 'this much', 'that much'. Every line here uses 이렇게 or 그렇게, and most are questions of disbelief aimed at the other person: 날 그렇게 싫어하니? (do you hate me that much?), 이걸 왜 이렇게 싫어하는 거야? (why does he hate this so much?). Note 싫어하다 is the verb 'to dislike'; the adjective 싫다 describes the speaker's own feeling instead.