While I don't really have anything against furries as a whole, the biggest problem I've personally experienced is that many of them have this bizarre and contradictory need for validation.
Firstly, they try to claim that anything even remotely associated with their fandom is somehow automatically a part of the fandom.
"Oh, you like cartoon animals and Disney movies? Well that means you're a furry too!"
No, that is now how it works! You are only a furry if you want to be, just like with any other fanbase. A lot of people, myself included, like anthropomorphic animals but have no interest in the furry fandom whatsoever. You can like something without having to be in some tangentially related fandom. Furries do not own the concept of anthropomorphism.
Secondly, these same kind of furries get really bent out of shape when their hobby is given even the tiniest bit of criticism. They claim that they're just having fun and don't care about any hate they get, yet then go out of their way just to defend themselves on any dissenting opinion. It's like they know it's weird, but they just can't accept it. It's a strange, conflicting, and yet desperate struggle for validation. It's rather pathetic, really.
Furries are weird, and that's okay. All fandoms are weird, honestly. The problem is when people make a fandom more about mob mentality and less about actually enjoying their hobby. That's why I don't subscribe to any fandom. Forget the labels, just like the things you like. Why do you have to assimilate everyone like the Borg? If someone doesn't want to be a furry then they're not a furry! Just leave it at that. They have their thing and you have yours.
But most of all, they really just need to come to terms with the fact that they will never be universally accepted. Your hobby is weird, and that is totally fine. Everybody has a hobby that others might think is weird. Just own it, for goodness sake!