In short: Availability.
I think Kdramas only started to be very big, internationally, in the last decade,* and it has something to do with the rise of streaming. Or rather, LEGAL streaming.
To give you some perspective, in the 2000s, when I started watching Japanese and South Korean dramas, they're mostly on level-playing field, availability-wise. You either watched them on terrestrial TV (if they're available at all in your country/local channel)--or you watched them on the internet (or downloaded them from the internet), with the help of fanmade subtitles.
The streaming sites have always been there, maybe from the mid-2000s, and always been not legal, but then somewhere along the way, came Dramafever** and Viki. And I guess it all snowballed from there. Now there's a safe and legal way to stream Kdramas, people's internet connection is getting better--making it possible to stream stuff without buffering, and as people's awareness of South Korea's cultural prowess grows, so is the popularity of their dramas (and movies, and music), I guess.
You might notice that I mentioned Japan early on. Now I think Dramafever and Viki could only afford to buy the rights for Kdramas in the first place, back in late 2000s--because they're relatively cheap, compared to Japanese dramas. Plus the Japanese are not really interested in exporting their cultural commodities, especially when they didn't/couldn't really see a market for them.
*I mean, I know that there's the so-called Hallyu, in the early 2000s, but I think it's mostly a "regional" phenomena, mostly in East/Southeast Asian and Middle Eastern countries.
**Early on, Dramafever actually snatched some fansubs for the shows that they had on their catalogue. As far as I remember, they didn't even ask permission. My point is, Dramafever actually started off as a pirate site, not really different from KissXXXX and Dxxxcool and their ilk.
I'm sorry that this is so long and convoluted. Feel free to ask or correct me if you see something that is not quite clear or accurate.