When you think of Korean names, the most common three names you can think about are Kim, Lee, and Park. It is very weird that only three surnames takes up almost half of whole Korean population. I don't know if any other country in the world have similar surname rates.
According to Korea National Statistics Office (KOSTAT), in year 2000, Kim(21.6%), Lee(14.8%), Park(8.5%) takes up about 45% of total Korean population. There are around 250 surnames in Korea, but no one really knows the exact reason why the three surnames take up so much portion. Yet there are several assumptions why the situation has happened.
First, the slaves from Chosun Dynasty got their first surnames inherited from their masters. In Chosun Dynasty, which is the last kingdom of Korean history, the royal surname was Lee(이, 李). Kim(김, 金) was the next authoritative, powerful royal family surname. As the dynasty fell, the slavery system also was collapsed, and the masters granted their surnames to the former-slaves. Think of the slaves in the US got their names from their masters after the Civil War.
Second, the common people purchased the royal names with money. At the end of Chosun Dynasty the social status system was on the verge of collapse. Some of the poor royal family could not afford to have slaves, and they are not respected. Some wealthy merchants had more power over the poor royal families. These wealthy common people even purchased the royal family's surname and made fake family tree as if they were born to be royal families. As the number of royal people increased, the Kim's and Lee's population also increased.
Third, the surname Park(박, 朴) is from Shila Dynasty, it is one of the oldest surnames of Korea. As the Shila Dynasty lasted for 1000 years, the descendents of Park spread out everywhere in Korea. Even though not as popular as Kim's and Lee's, Park's were also another common royal family surname in Korea.
Anyway, there is no clear reason why the three surnames are most popular surnames in Korea. Yet the three assumptions above might be the reasons why there are so many Kim, Lee, and Park.