First I will point out that turnout in South Korea in general is very high, for example, over 77% in the 2017 presidential election.
As for why the turnout in this particular off-year election was nonetheless remarkably high even by South Korean standards, all of the commentary I am seeing emphasizes the government's record on managing the pandemic and its efforts to contain the risks posed by voting. I've not seen anyone pointing to any other significant issue. In fact, support for the ruling Democratic Party was weaker before the pandemic. I don't find this surprising at all personally, but the professional analysts do seem a bit baffled in some cases.
To start with an early example before the results were in, from the AP:
While South Korea’s electorate is deeply divided along ideological and generational lines and regional loyalties, recent surveys showed growing support for Moon and his liberal party, reflecting public approval of an aggressive test-and-quarantine program credited with lowering fatality rates for the coronavirus compared to China, Europe and North America.
Analysts struggled to find explanations for the unexpectedly high turnout. Some simply gave up.
“Sorry, I really don’t have any theory for this,” said Yul Shin, a professor at Seoul’s Myongji University. “When turnouts are high, voters are usually trying to lay down judgment on a government that disappoints them. But the exit polls predict a crushing win for the ruling party.”
Analysts struggled to explain the surprisingly high turnout. Some said fear and alertness over the pandemic may have driven voters to support Moon’s government so it could fight the virus and its impact with more political stability.
Before the virus began absorbing public attention, Moon’s support was faltering over a decaying job market, corruption scandals surrounding key political allies and troubled ties with rival North Korea.
But surveys ahead of the polls indicated growing support, reflecting public approval of an aggressive test-and-quarantine program credited with lowering fatality rates for Covid-19 compared to China and some places in Europe and North America.
Voters CNN talked to ahead of the election were supportive of the decision to go ahead, and some said the pandemic made voting even more important.
Duyeon Kim, a senior advisor at non-profit International Crisis Group, said without the pandemic, it might have been a harder race for Moon's party. But during the campaign, the Democratic Party played up the government's response to the coronavirus.
"This tells other world leaders that how they respond to their own crisis could make or break their political fortunes because the pandemic is at the top of everyone's mind, most likely eclipsing other issues that would normally determine votes," Kim said.
I've yet to see any analyst with a fundamentally different narrative.