Was it just the dramas and movies or other way around? It's amazing seeing how many(?) fans of their dramas there is in the west.

Because of dramas. One day watching national TV I saw an episode of My Lovely Sam-Soon dubbed in Spanish. Since then I became very interested in dramas and of course their culture.
Dramas. In university, looking for a break, I stumbled across 1% of Anything. I was going through the whole "fake marriage turns true love" phase in romance stories and seeing it on screen was perfect for me. I was hooked since. And now I want to visit Incheon and Jejo-do lol. I even started to learn Hanguel.

Also, the reason I love Asian dramas and movies is the family aspect. They have a lot of respect and manners in their culture that is missing from the West. Plus, the emotionalism og their stories is not something Hollywood can easily capture. In my experience, emotionalism for Hollywood is that everyone sleeps with everyone else and break up each other's families and relationships. I don't like that. I like the 1 couple throughout the drama / movie who love and support each other in good times and bad. Asian media gives us that, and shows how respecting family members leads to a richer life.
Because of the beautiful language :) 
Because of Japan. I was first into jpop & jdramas and then got converted to kpop & kdramas \o/
For me it was a desire to learn another language, but I was very unsure which one to pick. The trigger then for me was that I was asked by a relative if I could locate a Korean drama on the net for her to watch, and when I watched it myself I decided that that was the language I wanted to learn. Now that I have numerous Korean friends, regularly visit Korea, and watch multiple Korean dramas, it's hard for me to imagine Korea not being a part of my life. 
For me it was the movie Seducing Mr Perfect. I was suffering from insomnia during my University days and decided to watch it on Youtube. i loved it so much that i wanted to watch more. I watched a couple more movies until I moved onto the dramas. I have never looked back since. 
manga to drama adaptations and then just branched out from there. BOF and Skip Beat
IU. I kinda just bumped into her music one day (2012 I think) and fell in love with her voice and the language.
Because of Arthur Golden's Memoirs of a Geisha - loved this book/movie and I was interested in geisha world and found information about Kisaeng and watched drama about them - Hwang Jini. Later I have noticed there are plenty of nice costume/historical dramas out there to watch - the genre I like the most :)
I liked Japan and somehow found out about Korean drama before I heard of Japanese drama. Then I fell in love with the Korean language and that was that. ;)
Haha, so many here went from Japan fans to Korea fans. :D Can relate except I went from Japan -> China. What you call obsessive fan of Korea? There's weeaboo for japanese. 
Honestly, I didn't know much about Korea at all. My brothers friend (now also my friend) was recruiting people to do summer camp in Korea, where the school would pay for our flight there and back. They were short on people, and so my brother asked me if I wanted to go, so I said yes. A few weeks before leaving for Korea, my friend suggested I watch a Korean drama before going, and after that I was hooked, lol. Going to Korea, made me even more interested in the culture. 
At first it was the dramas. My sister made me sit down and watch an episode that was in the middle of the drama, and i was so confused. So i went back and watched it from the beginning. Ive been hooked ever since. dang it sis. xD
I started from Jdrama to Kdrama. I'm all over the place, Cdrama, Thai Lakorns, I'm an addict and I love it!!!! ;)