After the lump passes through South Koreans will look around and see a lack of traffic congestion, abundance of housing, lack of competitors for their now abundant resources, and lack of disproportionately large population of unproductive elderly dependents.
Wanna guess what happens in that welcoming environment? My guess is people feel inclined to make some babies.
I don't understand why people tend to treat these low fertility rates as some kind of invariable biological dysfunction. These aren't infertile masses of people; they're perfectly capable of multiplying like rabbits.
The planet has shitloads of people, maybe we're finally reeling things in from an overshoot and are on the inevitably pendulum-like path towards population stability. There's no reason to panic unless you've got substantial evidence these people are physically incapable of having children.
Edit:
How about some DATA: go to https://population.io, select Rep. of Korea, plug in some age/gender b.s. and scroll down to "Age Distribution" at the left column. Now look to the left of ~20yo (younger than 20), it's a nearly horizontal line. What does that tell you? There's a uniform rate of Korean babies per year under 20 years old. To the right of this, there's a massive lump of elderly people with a slightly smaller lump of middle-aged people. There's no reason to believe this horizontal line left of 20yo won't just continue into the future; it looks quite stable for the last ~20 years.