It is very prestigious. When I first came to Korea as an adult, it was ranked #2. When I left, it was ranked #3. Either way, many people would literally kill to go there.
I went to Yonsei University Korean Language Institute and graduated. I also took a regular Yonsei University course (applicable to a bachelor's degree) in Korean literature and passed, no problem. I attended Yonsei classes of one type or another for three years (though with quite a few breaks, so only two continuous years).
The university is prestigious and it opens doors. I'm not even in Korea right now, but it helped me get a job here in Taiwan at a Korean international school.
It has a very large campus (especially for cramped little Korea) and good facilities.
By the way, it is hard for Koreans to get into, but not that hard for foreigners (unless you're Chinese, because there are so many Chinese students in Korea). Korea is trying to get more foreign students to study in Korea.
If you graduate Yonsei University Korean Language Institute, you get priority to their regular campus and also don't need to take any Korean language tests.
Entrance to Yonsei KLI is pretty easy and is a great gateway to studying at the regular university normally. Trouble is, you have to spend a year and a half learning Korean, and other countries often don't respect Korean degrees that much, so you could potentially spend 5.5 years in school and end up with a piece of paper that doesn't mean very much outside of Korea. That's why I ultimately decided, after graduating from Yonsei KLI, not to study normally at Yonsei -- just get a regular college degree from a US school instead. However, if you can sustain (for years and years) the motivation to attend a prestigious university in Korea, think about attending Yonsei KLI as a gateway to the regular university.
Attended Yonsei for two years, graduated from Yonsei University Korean Language Institute