There is no academic consensus on this question.
There are a lot of factors in growth that everyone knows about, but debates continue on their relative importance, directions of causality, time frame of their effects, and so forth.
Without taking any sides, I'll just list some of these factors here:
SK experienced high rates of investment. Some of this was FDI, but most of it was due to very high household savings rates. I know that the causes of high household savings rates is a highly debated topic with regards to China, I imagine the same applies to SK.
SK's initial conditions were actually quite favorable despite per capita GDP stats. Postwar SK's literacy rates for example were higher than substantially wealthier countries. Again, the causes for this are complex. Of course this involves government investment into education but behind that may lie possible cultural-historical reasons.
SK adopted export oriented growth that took advantage of its factor endowments (cheap labor). SK could export light industrial output that it had comparative advantage in and import high tech capital goods in exchange. The overall effect of this factor on overall growth and its causal relationship with other factors is disputed.
FDI it should be pointed out is not just for the physical capital. FDI is an immensely useful vehicle for knowledge diffusion. Investment into capital is a long run level effect, but knowledge accumulation is a long run growth effect. FDI is in practice inseparable from the transfer of managerial and technical expertise.
The role of the SK government is important for facilitating growth but again a lot of causal relationships are disputed. The state is needed to invest in human capital (healthcare and education), create institutional conditions where people are willing to save and invest and where markets can thrive. Also of possible great importance is the government's role in overcoming coordination failure, where the structure of a market of self interested actors can end up in a low growth equilibrium.
Just a sampling of various arguments from the academic literature:
http://documents.worldbank.org/curated/en/1993/09/698870/east-asian-miracle-economic-growth-public-policy-vol-1-2-main-report (argues for the importance of export orientation)
http://www.nber.org/papers/w4964.pdf (argues for the role of government coordination and favorable initial conditions)
https://www.foreignaffairs.com/articles/asia/1994-11-01/myth-asias-miracle (argues for the role of investment, without going into causality)
An excellent book that covers this period as well as many others (and ties them together into a surprisingly cohesive whole) is A History of Korea: From Antiquity to the Present by Michael Seth. It's a surprisingly good read if you're interested in the subject.
Short answer, post-Korean war investment, loans and other economic instruments provided Park Chung-hee (a strong U.S. backed dictator and father of the current president) the means to put in motion a semi-government run industrial king maker program that phased in core economic industries that provided a robust foundation for future economic development.
Longer (but not comprehensive) answer, Park Chung-hee's government was able to secure funding that enabled the development of a phased industrialization program mostly starting in the early 70s. Park's program focused on developing South Korea's Steel, Shipbuilding, Petrochemical and Automotive industries. A perfect storm of cheap capital investment and cheap and plentiful labor made Korea's early investments into these industries competitive in the regional and later global markets even if the quality wasn't as high other competitors. It's important to note that today Korea is the world's largest shipbuilder with the three largest ship manufacturing centers all in South Korea, and Korea is also 5th in the world in terms of vehicles manufactures and has major plays in petrochemicals. A side benefit is that all the expertise in construction has also made South Korea a major exporter of large project construction expertise (a great deal of Dubai was built by Korean construction firms for example).
Park implemented this plan by selecting various families (often with productive companies already underway in textiles or other low-grade manufacturing areas) and providing virtually free loans to them in exchange for carrying out his economic development program precisely as instructed. Several families didn't and their companies were virtually crushed afterwards, so it became a powerful incentive to follow the plan. Today, these companies are the major conglomerates (known as chaebol) with familiar names like Samsung, Hyundai, LG, etc. Other conglomerates known in Korea are SK, GS, Lotte, Hanwha, Hanjin, Doosan, Kumho and the list goes on. Today in Korea, conglomerate families form a kind of "royalty" and soap dramas often feature stories involving "millionaires" or heirs from rich families that are meant to represent conglomerate families.
In a sense it was a huge gamble. If Korea couldn't produce competitive companies quickly, the massive leverage in terms of outstanding loans would have bankrupted the country again. However, the Korean government was betting that cheap labor would produce the desired effect and it worked. The top 10 conglomerates grew at a rate of 27% a year during the 1970s.
In addition to this investment, the government invested massively in infrastructure and logistic enablers: roads, rail, rivers, canals, ports, etc were massively expanded to support raw materials coming into the country and finished goods leaving it. Today South Korea is one of the few countries in the world with a national highway system, high-speed national rail. Electrification, phone infrastructure (and then massive investments in now famous high-speed Internet) all followed or echoed these original developments.
At the same time, another parallel investment into education reform and (to use a direct word for it) propaganda began in order to turn a mostly agrarian and uneducated population into a manufacturing workforce. Free K-12 education became nationwide, the university system was greatly expanded and rural reform movements (saemaul) helped prepare rural agricultural areas for modernization.
The Saemaul movement, market price controls, fertilizers and other modern farming methods massively improved rural productivity and farming consolidation helped free up more work force to migrate to the cities and start manufacturing jobs. National telephone, electrification and television unified the country and made information available to rural areas for the first time.
Family planning became an important part of the Park administration's plan as well. Without the burden of many children, poor families could better afford to support the schooling of their fewer children, which better prepared them for jobs in the new industrialized world.
By the 1980s the economy had become large and complex enough, and sophisticated enough, that the old development model underwent revision. Several companies were forced to close to improve economic efficiency and to better focus the continued foreign investment. By this time, the educated workforce was sufficient that Korea was able to start another program, but this time in high-tech and electronics. In addition, changes in the energy markets created entry points into the American markets for a variety of Korean goods, including early Hyundai cars.
One of the keys to Korea's rapid success is a comprehensive STEM education and technology transfer program. The Korean school system places heavy emphasis on math and engineering topics to prepare a workforce. And then the Korean government or industry "buys" foreign goods under a license agreement wherein the items are licensed from the parent company and simply manufactured in Korea for the domestic market rather than ship them in. This has had the advantage of exposing Korean engineers to advanced foreign designed goods, everything from Japanese electronics to American Fighter Planes are produced this way (a topic for a whole other post).
Another thrust was on propaganda. Massive information programs were launched to encourage people to work hard and long hours, a relatively alien concept on a traditionally fairly laid back culture. Students were "encouraged" to study economically relevant fields, mostly STEM and finance, and "non-contributing" subjects were discouraged e.g. art, music etc.
The Korean model is often called "export driven development". Where a country produces and exports finished goods from raw product (it may need to import) in order to capture the value of the finishing processes. It's a model that works well in some cases (Hong Kong, Singapore, Taiwan, China), but has had less success in other countries that have tried it - mostly Latin American and African countries -- for reasons that are complex and often unclear, however, this is slowly changing. As part of Korea's diplomatic efforts, advisors are often sent to help provide policy guidance to poorer countries looking to model their own development plan. Here's a presentation from Japan that discusses some of these efforts coming out of successful East Asian countries. Look for "Korea's Knowledge Sharing Program" [Here's] (http://english.yonhapnews.co.kr/interview/2015/04/20/26/0800000000AEN20150420000200315F.html) an article about South Korea-Ethiopian economic development efforts.
It also has a tendency to stop working over time as the population becomes more affluent and thus labor (the core component of this model) becomes more expensive. This has happened in some sense to South Korea and they are now trying to start transitioning their economy into a modern services and trade-based economy. South Korea is putting massive investments to becoming a central hub for East-Asian trade and economic relations, for example the Incheon Free Economic Zone is one of the largest urbanization projects on the planet. This is of course helped by Korea's geographic position between Japan and China (2 of the 3 largest economies on the planet) and a reasonably short flight from Taiwan, Hong Kong, Shenzhen, Shanghai, Beijing, etc. Throughout history this geographic position has been the source of multiple invasions and conquest on the peninsula, now they're trying to turn it into an economic advantage.
edit
One thing I also forgot was a strong policy of domestic market protectionism. Until South Korea could develop a domestic industry that was at least regionally competitive, foreign players were virtually locked out of selling into the South Korean economy. For example, Japanese media, cars, electronics and video games have only recently started to be allowed to sell directly to South Koreans. Illegal video game arcades were actually quite a fixture in many areas during the 80s and 90s arcade booms. Korea also followed this same policy to help develop domestic film and music industries and today Korean films are highly regarded, Korean Dramas are highly popular from Turkey to Japan and K-Pop is a global export.
Edit Thanks gold!