- TL;DR: 하네요 does NOT mean "I'm just saying"; it is only pure coincidence that "I'm just saying" appeared in your translation at the given position. 하네요 is a particular form of the verb 하다 "do", and just like English "do", 하다 is used in hundreds of different ways, so uses of 하다/하네요/etc. can have wildly different meanings depending on where they appear. There is no one "matching English phrase".
Your subtitle is showing two (or three?) different people talking at the same time, so the sentences are mixed together. It should be understood as:
아이 진짜, 경찰 말도 무시하고 이제 의사말도 무시하네요.
Oh crap, [he is] ignoring what the cop says, and now [he is] also ignoring what the doctor says.
Person B (At the same time):
그게 아냐. 여기가 아파.
No, that's not true. It hurts here.
In Person B's speech, "그게 아냐" means literally "It is not (true)." The phrase is commonly used to mean "No, you have misunderstood; that's not true; It's not like that."
So, in this particular situation, B is probably trying to say something like "No, that's not true! It really hurts here. Believe me!"
So I think the translator inserted "I'm just saying" because it fits the situation and adding that makes the English dialogue sound more natural, but it does not correspond to any particular phrase in the Korean dialogue.
Same for "정말 날 깜짝 놀라게 하네요." It just means "It is really surprising me a lot." And 하네요 (or, at least some form of the verb 하다) is essential here, because it corresponds to "make" in "make me surprised = 놀라게 하다". Without it you have an incomplete sentence. I have no idea why the translator added "I'm just saying", but probably it corresponds to some phrase spoken before or after that Korean sentence.