My guess is it's pretty similar to drag hate.
Someone has a discomfort with blurring certain boundaries in society, and they express it as violent eliminationism — especially after being encouraged to do so as an "acceptable target" by bullies, propagandists, or other haters.
Just as with drag costuming & performance, furry is not itself a sex practice — but it's often closely connected with queer sexualities.
Furrydom also applies queerness to things that many folks feel protective of, such as cute animals, and characters that are (or appear to be) from children's media. That triggers the "queers are pedophiles" delusion, which is not new — look up Anita Bryant but has been heavily popularized by fascists of late.