The first part of your question is broader than Korea, so I'll talk about that first before getting into the region specifics of things.
Starcraft was an impactful game on several fronts. The most immediate after it's release was actually not so much it's gameplay but rather what led up to it. Starcraft was one of the first games to be specifically tracked by the gaming press as it made it's way through development, due to Blizzard's massive following from the PC gaming scene. It was announced at ETS in April 1996 and released in March 1998, with continuous revisions and updates spoken about publicly on the Battle.net forums. The Blizzard devs were very active in keeping up with their loyal following on the web, which really stoked the flames for the game to arrive.
When Starcraft landed, it was of course a great game, but it also riled a certain set of gamers for a company announcing a game so far in advance of it's release and seemingly delaying it needlessly. Take this editorial from PC Powerplay that notes Blizzard should be grateful for those who waited for Starcraft to come out (also note the mention of the infamously delayed Daikatana). I believe it to be true that Starcraft, for PC gamers, was at least one factor in the expectation that a "hype cycle" could last longer than one year. It wasn't to be expected anymore that a game you heard about recently was coming out soon. This might be positive or not, but it was an impact.
As to specific game mechanics, Starcraft did popularize the idea of asymmetrical strategy game opponents rather than relying on pure mirror matches or minor variations. Dawn of War is clearly influenced by Starcraft, as is Natural Selection, and of course the origins of DotA lie in a Starcraft mod. Indirectly this leads to the subtle differences in games like Counter Strike and overt difference in games like League of Legends. They didn't invent the rock-paper-scissors combo of units - wargames had been doing that for quite a while - but one of the first true instances of this in a visceral action game was in Starcraft.
Generally high profile games like this in history also raise the bar on technical aspects. Even Command and Conquer at this time had bad pathfinding bugs, and Starcraft made those unacceptable. The seamless multiplayer was part of Diablo (and to a degree Warcraft II) but true competitive multiplayer over an in-house supported service was pretty much exclusive to first-person shooters. Starcraft made it possible for high precision games to become viable via the internet.
That leads nicely into talking about Korea. There was no way to tell how popular Starcraft was going to be in Korea at first. There's only one truly good look at the Korean video game scene available in English that is useful to read, though it stops before Starcraft. Starcraft lands just a little bit after the American launch (earliest mention I've found is June 1998) and it quickly took off.
As the HG101 article talks about above, while there was a PC Gaming scene in Korea in the mid-90s, very few native Koreans actually owned PCs. In order to keep up on an economic scale with the rest of the world's computerization, the Korean government decided to invest in broadband internet technology. Between 1998 and 2001, over 8 million facilities were brought up to speeds of up to 1mbps which was well over the global average. This was of course not meant for games - mostly stock transactions and education - but it created a huge new opportunity for business in the public sector.
The Internet Cafe, or Game Bang, burst onto the scene to fill the economic gap. Usenet posts as early as 1996 speak of this growing cafe culture, wherein a proprietor would buy many PCs and allow people to use them for a limited time at a determined fee. Quickly this became the main spot to play PC games competitively. While it is certainly true that many such establishments would just pirate the games, the popularity skyrocketed for competitive games over these private broadband lines which led to more and more legitimate purchases.
By early 1999, the game and the culture attached to it had already drawn attention from the usual moral overseers but no real efforts were made to quash it. By May 1999, newspapers were reporting on competitive Starcraft teams. By October, the game had sold 1 million copies in Korea, bolstered by the multiple copies sold to Game Bang operators.
So this answers the question as to why, circumstantially, the game achieved it's dominance in the region. Right place, right time. There were other games though, some just as good as Starcraft and riding behind it in popularity even today like Counter Strike non-competitive games like Lineage. Unfortunately from a cultural perspective I can't really tell you why Starcraft remains king - or more accurately Brood War. Of course Blizzard games are just great in all and there's evidence that games like Command and Conquer held that sway prior to Starcraft which suggests something about the time and place for the RTS in general which made it appealing. I'll have to pass that on to people far more knowledgeable on the Korean psyche though.
Additional Sources:
Stay a While and Listen: Book I and Book II by David Craddock. A fascinating (if not complete) insight into Blizzard and Blizzard North from their origins to the collapse of Blizzard North in 2003. Development details on Starcraft as well as surviving evidence of Blizzard's Battle.net postings in updating development.
Video Games Around the World, edited by Mark J. P. Wolf, chapter on South Korea by Peichi Chung. It's far less history than market study (definitely not one of the best chapters in the book), but worth looking at for a little bit of additional info on this largely obscured side of gaming history.