I for one won't even touch a game if it doesn't have cute anime girls.
I've only played like 2 touhou games. I'm not a big fan of them, but I enjoy the fan art, and there's an endless array of sweet touhou remixes on Osu! I would even go so far as to say that all this fan-created content is a big part of touhou's popularity.
I think it's the music and the loli, really, and the fact that there's a girl for almost everything you can think of, including tons of animals, a bucket, shrine maidens, witches, gardeners, bands, and so on, so chances are you'll find at least one character you'll like. (If you absolutely hate women, there's Rinnosuke.) The music is popular to remix and to incorporate in medleys, and there's so much "art" of the girls. The games themselves are nothing spectacular; just standard danmaku games.
I liked Subterranean Animism the most. Imperishable Night was my first and probably had the most diverse gameplay. Wasn't a fan of the fighters or the Cirno game or the picture games.
Just look above you... If it's something that can be stopped, then just try to stop it!
Touhou isn't as popular as other games, it's partly well-known in the internet, but it hasn't transcended very much outside of it. Things like MLP have gotten a lot more popular.
To answer the question, I couldn't track when touhou became popular outside of japan, but the oldest gameplays I could find date from 2008, so I'm going to assume it was around that time, which is the year when SA was released, so obviously it's because everybody loves Satori. Still I don't think ZUN would have bothered releasing over 11 games if the series didn't get popular in Japan.
As for the reason why, the girls probably had their part and all, but there are so many games and anime with cute girls around so that alone couldn't make a popular game by itself.
Personally I'd say it's because of the challenge. The games themselves offer the player several difficulties to play from really easy to impossible lunatic. I once showed the game to one of my brother's friends who likes to play hard games, and he was impressed by the difficulty level in Lunatic that he downloaded the game himself and kept trying to be it (I should have showed him Kaizo instead). Orphen Nightford from here played DDC in his classroom once and all of his classmates were impressed by the difficulty and became fans of the game.
Another main reason is ZUN himself. The guy has been very open to the fanbase, he imposes almost no restrictions on fanwork and enjoys watching the fans creating doujin material, and all this fanwork also contributes to its popularity. Touhou is not just a bunch of shooting games with cute girls, it's also a lot of art, music remixes, fangames and more.
Basically everything that's been said here is a reason for Touhou's popularity. But in my opinion what's kept it so popular is the fans. Whether by fanart, fangames, or just playing the games, fans of Touhou keep doing what they do, and the games maintain their popularity as a result.
And with that, here's hoping the fandom stays going strong for years to come.