First of all:
Well yes, that's the question. Rather simple to be honest and there probably were many other threads such as this, but nevertheless I thought it might be nice to make another one, also I'd like to share with everyone why I personally like the series. This of course will be my personal opinion, so please no polemics, okay? =) Also,

It's actually quite curious that I like the Touhou series - I've never been a fan of shoot'em ups, even now. I usually prefer action/adventures with RPG-elements (DMC, Legend of Zelda, GoW, Metroid Prime series, you name it). Yet, in a very twisted way I highly enjoy playing Touhou games (in my case, reduce that to IN/PCB). Why is that? Or rather, why shouldn't I like these games, which is much easier to answer: I generally don't like hard games - call me a wimp, but when I play a game, I usually want to relax. I don't mind games being challenging, but at least for me IN/PCB are more than just challenging.
It's honestly rather nerve wrecking to play when you're not concentrated on the game, that is if you're actively flying into your own demise in form of coloratura little projectiles/lasers/knives/amulets and whatnot. And as I said before, usually I don't really want to concentrate to hard while playing, I'd rather just have fun with the game instead of focusing absolutely on it and paying dearly should my attention waver for just an instant.
Another reason is, that I'm probably just bad at shoot'em ups (hence I never really got into them). When I was young I sometimes played arcade games when I had holidays and therefore wasn't home (Where I live, arcade halls are exclusively for people aged 18+ or so) and I never was good at it - of course, as a child certain requirements just aren't that developed yet (hand-eye coordination, quick reflexes etc.), but still, I frankly just sucked. Years later, I'm still no good. I somehow manage to get through IN easy mode without continuing, can't do it in PCB easy or IN normal mode (damn you stage four). That might be because I just recently started playing these games (about.. 2 months ago or so) and I don't have all that much time for it, but I'm very sure that there are many people out there who could do a lot better.
In a nutshell, I suck at shoot'em ups, I don't really like them and I don't like hard games. So why the hell do I keep playing Touhou?
Frankly, I have no idea. Despite all of the above being true for me, I do like the games. Most probably it's because you can literally watch yourself progressing over time. A miserable example would be (but an example nonetheless) Kaguya's first impossible request, Dragon's necklace: When I first played IN, I couldn't not die to this spellcard even if my dear life depended on it. I just wasn't able to properly predict what exactly the lasers (or whatever those things are) were going to do. Today I'm somewhat confident at it (doesn't mean I don't die to it because of my lack of attention). Of course,it's easy and so on and so forth, but still, for me it was a small step which showed my improvement and the game.
There'd be other examples of which I could tell, but I'm quite sure everyone has their own personal 'milestones' (if you may call them like that)
where suddenly things which didn't work before then went pretty smoothly.
Reducing the series to gameplay mechanics (which is unfair, but still), that's more or less the only reason I can think of why I play them. But nevertheless, for exactly this reason I keep playing them. They're hard games and I keep getting frustrated at them, but then again when something suddenly works which didn't before, and be it just that I managed to capture a spellcard in practice mode after my 300th try (which happened..), it still shows me that I can indeed do it and it gives me the confidence, that I can do it again. So I keep on playing, no matter how frustrated I might get. I stop when I see there's just no point in it, that is when I clearly see that my concentration is just not there (that is if it's late), but otherwise - just keep going
The reasons above are now just focused on the gameplay aspects (which in my opinion is what I game should put its focus on), but of course there are other reasons why I like the series - Mostly the music. I daresay there's no need to elaborate on this point any further, the music is just great. Furthermore the series has a flair which makes it really charming. Ignoring the fact that there's only girls (and even that's not so bad), the settings are funny, giving a lot of room for inspiration (look at all this fanwork), the artwork is unique in its own way and generally, it's just very likable. It's hard to put it into words, so I'd better not try at all and leave it to you to try and understand what I mean.
So, this is why I like the Touhou series and play the games. Now I'm curious about you guys: Why do you like Touhou? (no need for such a wall of text, that's just a quirk of mine)
-Chronometer