Probable Explanation
Here's a
take on the fuller picture by a Korean Redditor (so take it with a grain of salt):
Quick rundown as a Korean: (South) Korea's two party system has been getting corrupted over time ever since its Democratic movement of the 80's.
Today is basically a power play between two corrupt party officials with Lee Jae-myung, the Democratic Party leader (questionable gang/mafia affiliation history, tons of people dead surrounding him, and potential North Korean connection) and Yoon Suk-yeol (current sitting president, conservative affiliation. dumb, impotent, potential connection to shaman who dictates policy for him, drunkard).
They've been in a hostile (competition) where one does shitty thing they other ends up doing shittier thing to nullify whatever the negative press may happen, all the while gatekeeping all the other legitimate proper candidates by sabotaging and sending them to jail. Yoon only became the president because Lee was also much more unpopular the last election.
But the scale has been tipping ever so slightly to Lee Jae-myung after the last set of corruption charges went not guilty verdict and his party was emboldened by that to create a set of laws that will basically make him bulletproof to all future charges.
Yoon panicked here and declared martial law to basically end Lee before Lee ends him.
This will not end well for Yoon since he does not have a popular support or the military, but Lee is basically a much worse version of Duterte so the hope is Yoon gets (rid of) Lee before he gets impeached himself.
Archive
To summarize, imagine President Yeltsin who listens to Rasputin trying to take down Lula da Silva (who has led Brazil to ruin, had corruption charges lobbied against him, but they were dropped thanks to a corrupt Supreme Court, and won election again). That is what is happening in South Korea.
Lolcow Nation
Democracy has pretty much never worked for South Korea. The country can be summed up as a de facto feudal oligarchic corporatocracy since its inception, ruled by noble families in the form of chaebols (named after the Japanese 'zaibatsu,' they are conglomerates that are inherited from the father to the children, i.e. Samsung, LG, Hyundai, etc.)
Edit:
Goddamn it, (possibly) got ninja-ed by
@1Tonka_Truck
Within the last five elected Korean Presidents (not including Acting President Hwang Kyo-ahn), three of them were charged of corruption, with two having served prison sentences and one of them killing himself "out of guilt."
2003-2008 | Roh Moo-hyun |  | Democratic (Liberal) | Killed himself due to impending bribery charges. |
2008-2013 | Lee Myung-bak |  | Saenuri (Conservative) | Went to prison for tax evasion, embezzlement, and bribery on a 17-year sentence. Only served 2 because of a pardon by Yoon. |
2013-2017 | Park Guen-hye |  | Saenuri (Conservative) | Impeached and went to prison for giving favors to a shaman aide and listening to her advice on a 25-year sentence. Only served 4 because of a pardon by Yoon. Also the daughter of military dictator Park Chung-hee. |
2017-2022 | Moon Jae-in |  | Democratic (Liberal) | Nothing yet... |
2022-present | Yoon Suk-yeol |  | People's Power (successor to Saenuri; Conservative) | Install martial law to overthrow his opponent after his opponent's party got away from corruption charges. May get impeached and imprisoned. |
If you want to go further, the legacy of Roh's predecessor Kim Dae-jung remained unscathed. Dae-jung's predecessor Kim Young-sam left the presidency at a 6% approval rate because of South Korea's failure to handle the 1997 Asian financial crisis and several man-made disasters. Within the last four presidents before Young-sam, three were military dictators or affiliated with the dictatorship (Roh Tae-woo, Chun Doo-hwan, Park Chung-hee), with the only non-dictator having been overthrown in a coup (Choi Kyu-hah).
Park Chung-hee overthrew Yun Po-sun in a military coup, and Po-sun was elected President after South Korea's 1st President Rhee Syngman lived in exile in Hawaii after the April Revolution.
So to summarize, out of South Korea's official 13 presidents, 4 were criminally charged and went through prison sentences (of which 2 were military dictators), 1 committed suicide, 1 got assassinated (Park Chung-hee), 2 got overthrown in a military coup, 1 lived in exile, 1 may be impeached and charged, 1 left an unpopular legacy, and 2 left an unscathed legacy.