There’s a number of factors that come into it but some of the more significant ones are the prevalence of PC bangs, the existence of baduk broadcasting on mainstream channels, and that conglomerates had a history of funding sports teams.
Bang or 방 in korean literally means “room”. PC Bang simply means a location where some businessman has set up a number of PCs and makes them available for hire. Similar set ups exist for other services. For example the ubiquitous “nore bang” (space added by me for clarity) literally means “song room” and it’s a place where you and your friends can sing together in privacy, a DVD bang is a place where you can watch a movie, etc.
Part of the reason why these rooms can exist is the relatively high population density, which impacts the environment in a number of ways. Dwellings are much more compact and closer together than you’d find in the suburbs of an american city, generally apartments vs stand-alone dwellings. That makes space for dedicated games or hobby rooms much rarer, but it also makes privacy (including from the rest of your family) dearer. Things like dedicated games or hobby rooms are generally not viable, and noise is an issue. Things like the “garage band” american concept of young people just jamming in their homes is much less viable in korea because you don’t have a garage and you’d be unduly disturbing your neighbours. The density also means more potential customers for any such facilities, people are much more likely to go to a PC bang if they can just walk there in 5 minutes vs needing to drive and park in order to get there.
PC bangs are available for any purpose that you might want to use a computer for but one of the things that their owners did in order to improve customer demand was to install games that you could play with your friends. In this manner it became an extension of the games arcade of yesteryear, a place where you can go and play a game for a relatively cheap price but for PC games. There’s also a social aspect to it just like there was with games arcades, with PC bangs encouraging gaming with friends. This social aspect also has a competitive aspect with certain people becoming known for being good at a certain game and attracting admiration. It’s the same way in which people known to be good at sports become popular. Basically PC bangs became the gathering point of people who were interested in the field. The people who were known to be good in the scene became the first recruits when the first professional teams were set up, which mirrors what’s been recently happening with NBA teams setting up e-sports teams and recruiting from the existing scene.
In other countries LAN parties mirrored PC bangs. The difference was that they were temporary and people brought their own equipment, but their reasons for existing was the same. People of like mind gathered together and competed together and against each other. A natural progression would have been for someone to create a permanent facility and infrastructure, i.e. a PC bang, but that never seems to have taken off.
When it became clear that there was an audience for computer game matches various companies started professional teams and recruited from the PC bang scene. Most professional sports teams in korea were born from conglomerates funding teams, for example LG Twins (baseball), Samsung Suwon Bluewings (football aka soccer), and Busan KT Sonicboom (basketball) founded by LG, Samsung, and KT respectively. Professional games teams were simply an extension of this approach and immediate profit would not have been the most important factor as it might be with amateurs trying to start a self-funded league. This is again similar to how individual NBA teams are now founding e-sports teams and how the NBA as a whole funds the WNBA.
Baduk, a strategic board game, has a long history of being broadcast on korean TV. High level players are able to become professionals and devote all their time to it via sponsorships and prize winnings at competitions. If you search youtube for phrases like “baduk korean TV” you can see examples. Baduk broadcasting became a template on which e-sports broadcasting was based, including the use of commentator teams and analytical discussion of matches, though obviously tailored for a younger audience. But overall it made it easier for broadcasters to accept that showing matches would attract an audience. It also helped greatly that Starcraft was such a well balanced game and it’s no surprise that it was one of the first major hits of the scene.
Overall there were a number of pre-existing conditions that made it easier for a professional gaming league to take hold in korea than elsewhere. PC bangs made it easier for a social gaming scene to take hold, the history of professional sports teams made it easier for conglomerates to fund e-sports teams, and broadcasters were already more predisposed towards broadcasting game matches.