Amazing Mooyoung had to leave because Yoil up close and personal was too much for her to handle.
The phrase is like clothes getting wet in drizzle. It's a Korean idiom for a person who changes in positive or negative ways over time without noticing there change, like how when standing in drizzle you don't realise your clothes are soaked because its light.
Althought in the chapter it references the song, It's a clever title referencing how much Yoil has changed as a person without even realising it. But since Soae pointed it out, she got her confidence back and realised how much she'd changed.
The sane with what Izuma said about hair, letting your hair down is the act of removing oppression and emotional baggage, in this case its based on a victorian custom, where in public woman would tie up their hair, but let it down in private as an act of freedom, to do it in public is showing no restraint.
Yoil was way cool in this chapter if she'd followed up with a kabedom Mooyoung might have fainted.