EDIT: as for why they're dancing. It doesn't need to make sense. It's IOSYS. You just have to look at other IOSYS videos to know that. It's just what that group does. For that one above the joke is merely that they're transposed Touhou characters into the 'ghetto' for a rap-battle dance-off. As for how it relates to Touhou, it's similar to the effect as if you animated various star wars character in a dance battle including Luke, Darth Vader, Boba Fett, C3PO. The point that they wouldn't be hanging out let alone doing that is kinda the point, but it would be easy to miss the point if you didn't know any of the source material.
I'll just leave this other video as a contribution:
https://www.youtube.com/watch?v=unRPhNH-YWoAlso here's a simple music video to remember who's who in Touhou. At least for the early games
https://www.youtube.com/watch?v=KqtdTdbSOEw----
Touhou fandom is basically a lot like Bay12forums. You've got Dwarf Fortress, that's the linchpin, but a lot of people here only dabble or haven't played those games in years. So, in theory, this place is all about and only about Dwarf Fortress. But it really isn't. It's about everything that grew up around that.
Simiarly, there are the Touhou games, but there is a ton of media surrounding it that has at times only a tenuous connection to the games. For example, there's a genre of touhou music, and touhou bands would be a subgenre of that, and touhou
metal bands is a subgenre of that, with different kinds of metal bands as subgenres of that. Pretty much the only Touhou stuff I consume is various Touhou metal music and occassional other styles of touhou band's music. Here's an entire last.fm category just for Touhou metal bands:
https://www.last.fm/tag/touhou+metal/artistsAlso a lot of memes about the thing aren't from the thing. So, someone referenced something from the game, and then there's a call-and-response culture where people make further stuff that references those memes, and in the end you've got memes in the culture that pretty much don't have any
direct link to the games themselves - fans have just decided to repeat that narrative enough so it's thing now. "Fanon" basically. By this point (which was about 10-12 years ago really) there were creators coming on board the Touhou thing who really never played the games in the first place. It's referencing the material, but purely as a sort of collective worldbuilding effort rather than directly referencing the games.
This is why it really
is like Bay12: There's this one guy who programs all the games, designs the characters, writes all the music, does all the in-game art, and self-releases. There's then the larger fan culture that remixes everything, decides they want to do Slayer-like covers of the songs, and creates all the fan art you'll see on Touhou videos.
None of that is official art, everything outside the games themselves is interpretation. This is helped because of Zun himself: he's a good composer, but not a great musician. He's a good character designer, but not a great artist. So for all the stuff he throws out, someone thinks "I could do that better" and they then interpret the music and art their own, high-fidelity way. Which has lead to hundreds if not thousands of professional artists having a go redrawing his characters. This kinda reminds of how Toady does Ascii then people come along and put graphics on it because they could do better. Zun also takes a very liberal view of copyright and fair use, allowing the community to do with it as they please. Zun is really the Toady of the Touhou community.