Other than sheer luck and enthusiasm?
When I applied, I'd just graduated with a BA in biology. I had about 1.5 years experience as a volunteer keeper at a big cat sanctuary, and around 10 years on and off as a small animal/pet exotics vet tech (not licensed). The keeper job was as a primate keeper, and I'd never worked with non-human primates before. Honestly I'm still not sure how I ended up getting hired. During my interview I was asked if I had any primate experience, and my lame answer was "...Um, you mean other than people? Nothing, but I'm willing to try."
From what I've learned since, it's actually fairly common for keepers to start off as volunteers at zoos/aquariums or exotics rescues, and many do have degrees in biology or zoology. If I'd known that at the time, maybe I'd have had more confidence during my interview.