Physical Danger
As we said, there is plenty of danger to this job. Cat scratches, dog bites, bear maulings—you need to be on your guard at all times. Having an adventurous streak will be an advantage. If you're the type who likes roller coasters and bungee jumping, you probably won't be quaking in your regulation boots when you have to save a group of babies from a grizzly.
There's also the ever-present danger of rabies. You'll get bit by lots of animals; if they're rabid, you'll have to undergo a series of painful shots so you don't get it too. It won't be fun, but just about anything is better than rabies.
The danger doesn't just come from animals, though. You'll have plenty of problematic people to deal with, too. There could be psychotic pet owners who freak out when you tell them they can't have five dogs, and there may be animal abusers who do things that are downright evil. Anyone who would hurt and maim a sweet little puppy is not someone you want to be caught in a dark alley with...but you won't have a choice in the matter. You're going to be chasing that person into that dark alley.
Some places will let you have a firearm or a taser to protect yourself; others won't. Either way, you'll have to be prepared.