The East Palace did some really deft character work to get us to believe that within days of meeting each other Saeng-gang and Gu-cheon were willing to risk literally everything for each other, even though neither of them (for very justifiable reasons) easily trusts other people and they start out being very annoyed at each other.
Gu-cheon is obviously and reasonably mad at the whole situation, and when he realizes Saeng-gang is a princess, he probably takes it to mean she's there primarily to monitor him (which is indeed the task the King gave her) and not so much to help him. Hence it's easy for him to take it out on her and basically try to brush her away as much as he can.
Meanwhile for Saeng-gang being able to come back to the palace is a chance to find out what exactly happened to her mother, and meeting Gu-cheon she in turn is frustrated since she realizes his attitude will make it all the more difficult to achieve what she wants and to possibly be able to stay in the palace for longer term. But she is stubborn, so she keeps asking questions even though he barely answers, fetches him the peach tree branches even though he doesn't explain why they're needed, and when she writes that first letter to the king she throws it away and never sends it, because no matter how frustrating she finds the situation, he's still the only way forward for her.
I think Gu-cheon despite his misgiving starts to warm up to her pretty much immediately, probably partly for the simple fact that she is genuinely paying attention to him, is interested in him, and after his mother's death there probably haven't been really anyone to do that. And he does on that very first day start to recognize she actually does mean to help him with his task. Hence, when she follows him to the pond despite knowing it might be dangerous, he allows himself to trust her enough to have her help him by letting her play the bait (although not telling her that's what it is) and taking her bell as a talisman. Then, when she does what he instructed and successfully distracts the spirit, and later saves him by pulling him back to the real world, he starts to respect her.
For Saeng-gang, seeing what he has to do, what he is being made to endure, is the turning point, and from that point on she gains a better understanding and respect for him.
The conversation after Gu-cheon has returned from the spirit world for the first time is an important turning point, because they both choose to be honest when they wouldn't necessarily have to. Saeng-gang tells him she regrets what she thought of him before, even though she could have just quietly adjusted her behavior going forward. In turn Gu-cheon lets her know he knows she's the king's daughter, even though he could have kept it to himself, but he now trusts her enough that he doesn't have to be so guarded about it. This leads them to agreeing to work together not only on the matter of the spirit but also Saeng-gang's investigation to her mother's death, and her telling him she can hear the spirits. It's not immediately smooth sailing, as it shouldn't, they still butt heads over things, and need time to learn to properly communicate with each other, but it's a start.
Another important point is when they're discussing the spirits of the salt store house, and he casually says well done for finding out, and she's very taken aback by it, mentioning he's the first person to find the ability to hear spirits valuable. More than that, I suspect that in general not that many people (if anyone) has praised her for her skills and abilities since she left the palace, so being able to do something that's genuinely useful must be important to her. That's also when he starts to understand more of her background and find compassion for her, as well as parallels about their situations despite the difference in their birth status.
Then there's the argument they get into before he goes to see the spirit of the guard, where he accuses her of being just like the other members of royalty, just using people for their own benefit. And she doesn't deny it, because there's a grain of truth in it, she needs him to reach her own goals and isn't willing to jeopardize it. However, unlike the king (who on many occasions demands things that are detrimental to the task, because he has no interest of understanding how things work with the spirits), she does her best to understand what he needs and how to help him, because to her it goes both ways; if she expects him to help her, then she has to in turn help him the best she can.
Imo it's these two things, her sense of responsibility for him and the connection where he seems to be the only person to actually understand the part of her that everyone else have for most of her life pretended didn't exist at all that pushes her to put aside her fear (and she really is terrified at that moment when she's preparing to go under the surface) and follow after him to bring him back. He has become that important to her.
For Gu-cheon the fact that she did come after him is what completely changes things. He's mad at her at first, but it's really because he was frightened, but when she explains that she was terrified and still did it he seems to realize what it truly meant. She actively chose to risk her life for him, when no one else in the world would, when in fact most people are happy to have him risk his life for their purpose, and even his mother who should have always protected him nearly killed him. At the start of the story I'm sure he thinks there's no one in the world who ever would risk themselves for him, but Saeng-gang does only days after first meeting him. That's why from that moment forward, to him it's unacceptable that anything should happen to her, and that's why he has no option but to stay even though she tells him to run away.
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