Perhaps because the laws originate in military dictatorships:
the South Korea government engages in active Internet censorship based on three laws: the Nation Security Law , the Basic Press Act , and Article 21 [of the 1987 Constitution].
Democracy or freedom of speech doesn't have deep roots in South Korea:
Censorship was almost forgotten under the country’s liberal governments between 1998 and 2008, but critics say the country regressed under the conservative government of Lee Myung-bak and the problem has been exacerbated under Ms Park.
So the laws that enable it in broad terms apparently weren't repealed, even in the more liberal decade. Which made their stricter re-enforcement in more recent times a fairly simple matter.
A fairly similar view is expressed in a much longer academic article:
Many of the laws that provide a mandate for this censorship were written
in the 1990s, while others have their origins in pre-democratic Korea. They
impose broad restrictions on expression in the areas of obscenity, national
security threats, threats to public order, and political debate. Though
controversial, they have endured through governments of both the Left and
the Right. Some scholars explain their persistence by reference to recurrent
defects in South Korean politics, including regionalism, ideological
polarization, corruption, weak political parties, and personalistic rule by
presidents. Regardless of the explanation, commentators and scholars
interpret the continued existence of these laws in two ways – some see
them as holdovers from Korea’s authoritarian past that will fall as its
democratic institutions strengthen, while others see them as evidence that
South Korea is developing a new model of illiberal democracy in which
the organs of the State will extensively dictate what individuals and civil
society groups may or may not say.
That article also develops interesting parallels with Chinese censorship, which I'll omit here for brevity. It also discusses the role of the (post-1987) Constitutional Court in both limiting but also perpetuating some forms of censorship in South Korea, through their various decisions interpreting the [fairly] old laws.