When Lee Eunmi approached her with the offer of going outside with her, Moon Jiwoo had no reason to decline the female. Staying inside of the palace felt too stuffed to her, and regardless of how large the place was, going outside seemed like a really good option to her.
It would be a breath of fresh water to Moon Jiwoo.
Although Moon Jiwoo had given Sookyung her word on how she will not be trying to escape from the place, he still doesn't trust her enough to let her out of the mansion.
Seeing that Eunmi was presenting her with such an opportunity, she was exhilarated. Moreover, now, rather than being alone, she had someone to hang out with when she was outside.
Certainly, Oh Sookyung wasn't ready to let her go out yet, and that's why, Eunmi decided to sneak out of the palace with her. Although Jiwoo had no clue on the female will be doing so, she trusted her with it.