In episode 19 when he is telling her the 3 conditions, he asks her, "Who is the most important to you?" and she answers "Me." As the eldest daughter with all the responsibilities on her shoulder, the only person Bora could ever trust is herself. She is not allowed to put her faith in anybody else and burden them to solve her problems because she needs to hold in her pain and suffering for the good of her younger siblings. That's why when Sunwoo says that I want to be an important aspect of your life that you can't just remove when things get tough, he is asking to be seen as somebody who is irreplaceable and who she can share her burdens with, depend on when things get hard, and lean on to help her in her times of pain.