I'm wondering why ya'll think Asian dramas (apparently mostly Korean dramas) are so special. I tried watching "Love from stars" which got huge raving going on and it wasn't anything special. I'm mostly "into" these for sake of language studies but don't think they are anything special actually. 
It's a matter of taste actually; I personally started watching Japanese dramas ever since I was a little girl so it became a normal thing.
I think Asian dramas' lovers got tired of the western way of doing shows so they decided to change the air a bit. Some of them even prefer Asian dramas over other shows.
Well, for me, I'd say that a major thing is that there is usually just one season. I don't like episodic shows nor shows that will just keep spinning on with endless seasons. So a Korean drama will have that one major story line and after its done it's done. Time to move on.
As far as I can tell.. they seem to have the same exact amount of crazy overly-dramatic writing, just these come with added bonus of 10x more ~cuuteness / dorkiness / etc to make it more bearable, at least in the ones I've watched so far, lol xD (admittedly mostly "love stories")

I personally prefer episodic shows that I can "grow with" for years over time (I think 20ish per season, ie Sept thru May "school-year" / US style, is a good number, or the amount just under that btwn 10-16ish over a couple months here & there for cable & most other non-US series, etc), but most of them I never really want to actually marathon- they're good in their small-but-long-lasting doses, one at a time, spread out over months.

Then there's these "shows", which are basically like reeeally long movies that I somehow can't get enough of, haha. And while I love movies, them ending too soon was always a bother- getting addicted to chars whom I'm only going to enjoy for less than 2hrs & never see again? What a tease! This way, I get to have many, many hours with certain chars/actors & at least ~length-wise, it's kind of like reading a book, in that they are so flexible (as opposed to most of my procedurals which are self-contained stories in every ep, even with seasonal arcs, etc) & easy to consume at one's own pace.
I think the reason they haven't really appealed to you yet is because you just haven't found the right ones. My Love From the Stars in my opinion is terrible. I think you need to find the dramas that aren't as popular. What genres do you like? Or what are some of your favorite western shows? Maybe I could find something similar. Going from your plan to watch list, I would say watch MARS. It's actually one of my all time favorites.

The main reason I like asian dramas is like Fira said, they are typically only one season. I like my shows short, sweet, and to the point. If a show goes on more than two seasons, I'll lose interest. I also feel like asian shows have morals where western shows don't. That and the romance is sweeter in asian dramas, they don't sleep with each other the very first episode.

Edit: I noticed you have an mal, have you tried watching any animes that got turned into dramas? It's just a thought. That's how I originally got into dramas. I was first an anime fan.
I like them because they have endings. They are a whole story in twenty episodes. It is like reading a paperback romance that lasts for twenty hours instead of two or three.
One of the main reasons I prefer asian dramas over wester is the same as Goldenseal.

There aren't too many western shows I was able to finish, and those I did generally always had at least 1 horrible season (that's over 20 episodes!!). I spent 7 years watching Bones and Supernatural. I loved the chemistry the leads had in bones, but the just kept dragging it, and their story just stopped making sense to me. As for supernatural, it's the same story that keeps on repeating. I can see why they are dragging it, the lead guys are gorgeous and amazing in their roles, but the show stopped being creative quite a long while ago. Prison break should've ended at season 2. I HATED season 4 as a whole. Many shows I've watching one season and just don't have the energy to watch more.

My preference in Japanese dramas, which generally are pretty creative, so I find things that is hard to find in the west, or anywhere else for that matter (And I have seen dramas and movies from many different places!!).

I think for many, it is more the way relationships are presented in asian dramas that attracts them. 

In any case you can't just watch the mainstream dramas and decide there isn't anything special about them. There definitely is something special about them. In fact each country has it's own flavour. Those into asian dramas most have also been into other types, and pretty much know what's out there. You would have to explore this world deeper to decide if there is something special about them or not. In the end, none my be up to your taste. Or maybe you'll find something special about them and and get "into" it :P 
I think novelty is a big factor. Also, the Asian culture is quite different from the western one, so some people might enjoy that more. 
The INNOCENCE lol in american and spanish dramas u see sorry but S** in EVERY EPISODE!!! It's disgusting, nasty and wanna vomit. 
I think vampire diares will get AIDS soon ( classmate told me how many times they all switched ! ) 
Western dramas have more violence, sexuality and promiscuity. I love Asian dramas because of innocence, fairy tale romance, fun factor and laughter. They are definitely lighter and appropriate for younger generation too.  I like the plots better. They are shorter in length and they are addictive too. I would say, they are magical, many times unreal and ridiculous, but I like that. I like to dream, and that is what relaxes me lately. We live in violent world today, and Asian dramas keep us away from that and move us to places when life looks that much more beautiful. Also, most of the guys are portrayed in a good way, the way women would like to have them - caring, loving and romantic ....So why do I like Asian Dramas so much -  did I say I am a big romantic at heart, and that is what gets me each and every time in those dramas??? LOL
For me, just as others have said, it partly has to do with the length. I dropped a number of American shows because they're getting completely ridiculous (Bones, Pretty Little Liars), at least with Asian dramas I know I won't have to torture myself for long to find out how it ends. Also if it ends terrible I won't feel like I've wasted years on something that was only going to disappoint me (How I Me Your Mother). I also feel it's easier to relate to the characters in Asian dramas as well, their personalities, family life etc.
Just as how I don't like one Asian country's drama more than the other, I don't like Asian dramas more than Western ones or vice versa. When push comes to shove, I think they all have their strengths and weaknesses. I think some Asian countries do certain genres better than their regional counterparts and I think Western dramas are to some extent better conceived and executed, story wise, while Asian dramas will have superior characterizations that feel more authentic and human. But it's not a perfect world in either ... while I agree with those who complain that Western TV tend to be too overly sexualized and violent, I think that it's the opposite extreme (too innocent and prudish for shows targeted at grown adults) in Asian dramas. I care for neither extremes. 

Western and Asian dramas -, as are the dramas from the other Asian countries, all have their flavour and it depends on what I have a taste for at the present time. I've been watching Western dramas all my life. I still watch and enjoy them, though I've dedicated most of my drama watching leisure time to Asian dramas the last 2.5 yrs. This is simply because Asian cinema & TV present a fascinating new world, which I previously had very little knowledge of and personal interest in. However, now, I've had my prejudices and preconceived notions challenged and corrected, and certain beliefs confirmed and reinforced. I've challenged myself to learn a new language, and I get to understand a new way of thinking and looking at the world through unique storytelling & awesome history (every time I watch a sageuk). But perhaps most importantly, I meet characters that feel more relateable to me, as if they walked right out of my life. These are some of the reasons I watch Asian dramas. 

I don't see them as more special than Western or any other drama, just different. I like variety, therefore, I am not among those comrades who religiously stick to one country's dramas. I watch shows from all over the world as long there isn't some sort of cultural dissonance which renders them unwatchable. I assume in time the craving for each flavour -- Western vs Asian -- will balance out and I'll watch both equally. I kind of have the same attitude towards genres. I'll watch just about anything, but I go through blocks of time -- some long, some short -- in which I only watch a certain genre or genres. Doesn't mean I don't like the others, I'm just not in the mood for them right now.
I like that they have a limited run and don't drag on for years getting boring and stale and burning out before being canceled. I feel like there is typically a lot of growth and development in the characters in Asian dramas; more so than in Western media.

I still watch Western TV sometimes. I don't necessarily think one is that much better than the other, it's just nice to have choices.
Others have mentioned it, but the big thing for me is that they are more concise. For example in American Soap Operas, many of these have been going on for over 50 years... And really have almost no plot (in my opinion.). Asian Dramas tell a complete story in 12 to 40 episodes and then it's done. There are a few asian drama that have hundreds of episodes but this is the vast minority. I appreciate that they can tell a story quickly, and capture my attention long enough to tell the story and then let me move on, without needing years/decades to tell a story.

I also feel that Asian drama has more feeling. I cry ALL the time during them. Like hyperventilating even. I can't say the same for many American films or series. Everyone I know raves about "The fault of our stars" (an american drama about two young lovers who are fighting cancer). I felt nothing at all watching this film. And I'm just thinking to myself, these people have never seen a sad movie, they need to watch some Kdramas lol!

The last reason I prefer Kdrama is because I find the men more attractive, that's just me though :) Eyecandy is a good thing lol. But not just physical beauty... their actions and facial expressions and just everything, I find them so cute :). I think it's mostly the way they smile and their eyes. I know South Korea is like the plastic surgery capital of the world, and that it's not very realistic, but still, if I'm gonna watch someone on TV I want to see someone attractive lol.
Most Asian dramas are more innocent and...clean I guess? Not ALL I mean but usually, Western dramas are more sexual and tend to be more adult-ish than Asian ones. And when there is romance as a main factor in Western dramas, there is hardly any comedy which is, in my case, a big turn-off unless it's a melodrama. That is an exception of course. So, they tend to focus too intimately on romance and that makes it too intense for my liking.

Oh! Asian dramas make me squeal and fangirl over the couple since they plan out the story well and there is a development of character. It doesn't make me cringe when the romance is too sudden and based off looks.

And I'm not a big fan of My Love From Another Star as well. I guess it's just our preference.