My answer may not be that conventional, but it was "First Love of a Royal Prince"...probably most of you haven't heard of that before. I remember it because my mum was watching it dubbed over with Chinese, and I always wondered why their mouths never matched up with their voices lol. The drama, thinking now in hindsight wasn't that amazing at all, however at the time it just seemed so interesting; I had never seen anything similar to it before (By the way, this was probably around 2005-2006. The drama wasn't that popular when it first came out, but got a lot of publicity around a year later after it spread across Asia)
And so from then on, I went with my mum occasionally to buy new dramas at a local newsagency (of the time, and they were always dubbed over) and that's how I kinda got into K-dramas. However, it wasn't until years later (around 2007-2008) when I had free access to the internet at home that I started watching dramas more frequently.