I agree with everything Glasso said, and if I might add to his excellent points. While I love Korean, Japanese, Chinese and most other varieties of Asian drama, action, and even some of the comedies, one of the main reasons I watch is featured in your picture there. Ms. Song Hye Kyo makes almost any plot line entertaining.
Seriously though, some of my favorite movies are Thai and Korean westerns. The Good The Bad and The Weird is a good example of this genera. The Thai ones are especially great because they cheerfully mix traditional Thai magic with western themes, scenery and costumes. There was an immensely popular Mini Series which I watched two years ago when I was in Bangkok that I really wish I could remember the name of. If all the actors weren't Thai, it could have been a particular bizarre season of Bonanza.
But, back to your question, Part of the reason I think the Korean and other Asian dramas are so popular is because they are not caught up in the obligatory plot lines that so much of American Tv and Movies follow. In a good Korean drama it is entirely possible for the movie to end with the hero in abject misery. Hell, in Japanese drama it is almost expected. To me, it makes the whole thing seem more real. I am constantly on the edge of my seat because I really don't know what will happen. It really isn't possible to predict which characters will make it the entire season and live happily ever after. The story lines are written in such a way that the audience is constantly left guessing what will happen next.