According to the book Soccernomics by Simon Kuper and Stefan Szymanski, there are three factors which predict the relative success of a national soccer team --- GDP per capita, population size, and the experience of the national soccer program. Right now, South Korea does well in the first couple categories. It's not bad in the third one either, though I have some ideas as to how this could be improved.
PoD: After seeing the success of the 1964 Summer Olympics in Tokyo, Park Chung-hee decides that South Korea needs to host a major sports tournament to raise its international profile. He gets the Korean FA to launch a bid for the 1982 World Cup, modeled after the failed Japan 1970 bid, but with some improvements.
Now, in 1966, South Korea is pretty damn poor still, and viewed as a long-shot bid. IOTL, West Germany and Spain agreed to support each others bids for 1974 and 1982, all but sealing their respective hosting right. ITTL, for whatever reason, they don't reach such an agreement. West Germany wins the narrowly wins the right to host 1974, Argentina 1978 IOTL, and 1982 is hotly contested without West Germany supporting Spain. Without enough support, Spain supports South Korea over the Italian bid, with the assumption that South Korea will not actually be able to host the cup. South Korea is awarded 1982 to a collective reaction of WTF.
But Park, who avoids assassination ITTL, is pretty for serious about all of this. He invests serious money in the national soccer program, allowing South Korea to qualify in both 1974 and 1978 (though they don't exactly do that well). By 1982, with the international spotlight on South Korea, there is a wave of student protests calling for democratization. The first democratic elections are held in 1981, which Park wins handily over controversial opposition leader Kim Dae-jung. The 1982 World Cup goes surprisingly well. Hosts South Korea make it to the quarterfinals on the backs of some questionable referee decisions. The profile of the sport has been raised to astronomical levels in Korea, as well as the rest of East Asia.
Fast forward to 2015. President Roh Moo-hyun, Vice President Kim Jong-un, and Prime Minister Park Geun-hye are all in attendance in Brazil when the Unified Republic of Korea wins their first World Cup. Having kickstarted their soccer program a generation earlier than OTL, Korea has the best men's national team in AFC, and one of the best in the world. The success of the 1982 World Cup opened the door for East Asian soccer too, allowing Japan to host a cup in 1998, and China in 2010. Australia and Vietnam want their turns next, though Korea is planning on launching an audacious bid of their own in the near future. Most of Korea's top players play in Europe, though the K-League is much stronger than OTL too, filled with old European and South American stars eager to retire to glamorous Korea, and young Korean talents from all over the peninsula, eager to become huge stars in their soccer-crazy country.