K-dramas tend to have consistent character types and recurring tropes such as the poor female lead (who exclusively owns designer clothes) whose parents run a fried chicken shop and the future CEO heir to the company male lead. Recurring K-drama characters will be the rich and popular female bully, the side-lined second male love interest (who is always the better choice) and the single parent of the female lead who is absent due to debt, an affair, etc. These easy to identify characters and storylines were intended for Korean housewives to be able to follow while going about housework, and the drama of the repeated shot from a different angle and the dramatic romantic build up created popularity.
The link between K-dramas, K-pop and Western popularity is propped up by the piqued interest in Korean culture and language
Recently, with the rise of K-pop, K-dramas have reached an overseas audience. I would say the first K-drama to do this was Hwarang: The Poet Warrior Youth! (2016) in which V from BTS had a minor role before being dramatically killed off. Since then, adaptations of Webtoons have been increasingly popular such as True Beauty (2020) (ugh) which was somehow a hit. Popularity only grew when K-pop idols decided to try their hands at acting such as IU, Chuu (Loona) and Cha Eunwoo (ASTRO) as well as K-pop groups and soloists appearing on OSTs. The link between K-dramas, K-pop and Western popularity is propped up by the piqued interest in Korean culture and language, with K-dramas being a microcosm of Korean culture that is accessible by Western audiences.
K-dramas have an interesting production history as before the 2000s, they were made in-house by the TV channels themselves. There is also usually only one director and one screenwriter for a K-drama however, this quirk has been lost in recent years. This is due to the rise in Netflix original K-dramas meaning that those shown on Korean TV channels domestically are less internationally popular.