As others have said, furries are anthropomorphic animals (think StarFox, Zootopia, etc). People will create original characters (OCs) of humanoid-animals, and fursonas, which is a representation of themselves, as an animal. Some people make/commission costumes (called fursuits) of said characters. They tend to get a lot of hate because there is a HUGE sexual connotation to them. It's more than just the sexual content than people associate it with, but the furry community does come off as "loud and proud" about that side of things, more so than other fandoms, so that's kind of what people know about it who aren't involved. It's mainly just a creative outlet for people, and allows people to be who they want to be or to express themselves in a different way.
Anyways, the whole explosion of furry art on Splatoon happened when someone posted in the Miiverse with some furry hate (something along the lines of "I hate furries/delete all furries" or something of the sorts). Then someone and their friends started to spread furry art (you may have seen the drawing of the dog with the flag "Draw more furries"- this is the one that started it), and it just kind of blew up. There's an article on Kotaku about it, if you want to read more about it.