Touhou has innumerable doujins/fanfics detailing the Back Story of the characters, since not many characters have one to begin with, and even if they do it's still pretty vague. By far the two most common recipients are Rumia (so popular that any fan referring to "EX-Rumia" is immediately understood) and Sakuya (serial killer, former monster hunter, or Lunarian being the three main theories).
Touhou Tonari and Though the Wind Cries are good examples of this.
Detailing the future of characters is also common, as amongst the incredibly long-lived youkai there are a few humans that will die relatively soon. Reimu, Marisa and especially Sakuya (again) are the targets, though Sanae became increasingly popular after her arrival.
The "Outsider Fic" subgenre, where somebody (often an Author Avatar, but may also be any kind of original character) somehow ends up within Gensokyo, either by accidentally/purposefully slipping through the Hakurei border (which, according to canon, is common), or by Yukari (in a sub-subgenre increasingly known as "Gappy Fics"). As a result, Fish out of Water antics ensue as they get acquainted with this Fantasy Kitchen Sink, and the outsider must either learn to live in this new land or find a way back home. Occasionally, they'll arrive just as an incident begins and they're forced to take part in it for one reason or another, teaming up with the canon characters. A fine example of this is ChibiChen's Mind The Gap.
The four great Fandom-Specific Explanations of Touhou are summed up by a well-known meme, namely this doujin panel◊. (Which has actually been altered from the original, which just had the first three lines.)
Magic: Magic in Touhou is not overly defined, so it can be used to justify whatever need be justified.
Eirin's shady new drug: Eirin's tendency to make unusual concoctions of dubious pharmaceutical benefit and bizarre effect is often used to justify strange happenings. Inaba of the Moon and Inaba of the Earth, generally considered canon, uses this one.
It's a Moriya Shrine conspiracy: After their introduction in Mountain of Faith, every single "main" game in the series (Subterranean Animism, Unidentified Fantastic Object, and Ten Desires) was either directly or indirectly caused by the Moriya trio's (more specifically Kanako's) attempts to gather more faith from the population. Suffice to say fanon took it and ran with it.
Templates like that are forbidden: Marisa says this in response to the above list. In more comedic stories, this has occasionally been used as an actual excuse for the plot.