I'm not going to bore you with a million details but here's a few reasons.
1) She's extremely realistic and easy to relate to. Don't get me wrong the entire mane six is but she pretty much nails the whole "introvert nerd" without it being her whole character.
2) She's adorable as hell. Granted Fluttershy was meant to be the embodiment of cute but personally I think Twilight is cuter, probably because it's not her only character trait and she doesn't even try to be.
3) She's actually quite funny. Sure compared to Pinkie Pie her humor is more subtle. Like in "Pinkie Keen" when before she went on a mini-rant about magic she climbed on a soapbox. lol, and let's not forget when she has her little break downs.
4) She reminds me of myself, we're both anti-social, quirky, likes to read, prone to be OCD over many things, if you know me in real life you wouldn't disagree. Hell she even looks like me a little, lol
5) I enjoy shipping her with every living thing on the show, I know Rainbow Dash is usually everyone's "black dress" but Twilight is my favorite character to ship. She's just so...versatile.
Those aren't the only reasons but like I said I don't want to ramble on lol
Her intelligence. She's independent and clever, and I love her quick wits and sharp resourcefulness. Twilight's very strong mind has helped the ponies many times, and I'm sure it'll continue to help them throughout My Little Pony: Friendship is Magic.
I too can relate to her a great deal. Like everypony else, she is complex and well-developed rather than being mono-dimensional and limited like so many other characters from other things. Twilight's type of sarcasm and dry wit are to die for! She's a pretty pony, and I just love her bouts of craziness, her OCD, her smarts, and her rational, scientific mindset.
I also love and relate to her aptitude for reading and research, her squeeful glee over the little things that excite her, her naturally organized nature and leadership skills, etc. Basically, every detail about her is pretty awesome or serves to make her interesting. Her outstanding magical talent, amazing assortment of associates (both equine and otherwise), the way she's never been too sociable...her courage and determination to face down any enemy and do whatever needs to be done in any given situation...her bookish, studious nerdiness...!
She actually IS quite versatile shipping-wise, too, lol.