The answer, as far as I can tell, is to be a crazy, reckless sonofabitch who likes animals.
I downloaded a copy of Richardson's book, Part of the Pride: My Life Among the Big Cats of Africa, to give it a read and see if I could gain some insight into Richardson's relationships with the lions and hyenas that we see in the video. What I didn't realize is that I've actually seen Richardson on TV before, when Animal Planet was still pretending to be a channel that featured animals instead of Sasquatch and hillbillies. He featured prominently in an episode of Growing Up, when he was working to hand-raise a hyena pup called Homer. You can view the whole episode here, but I feel I do have to spoil you regarding one thing: Homer unfortunately passes away.
It's one of the things Richardson writes about, roughly two-thirds the way through the book. As a writer, Richardson makes a good Lion Whisperer. He wrote it with Tony Park, who I think is primarily responsible for presenting Richardson's thoughts in a relatively logical format. The reason I call him crazy and reckless is that he freely admits that he is - when he's not rolling around in the grass with lions and hyenas, he's riding motorcycles and flying ultralight aircraft.
Richardson's approach to his autobiography is, I imagine, very similar to the way he approaches his life and work - fully in the moment, full of emotion and curiosity and deep interest. He writes about growing up being fascinated by keeping and raising animals, and he puts a strong emphasis on building relationships with animals. He carries this importance on animal relationships over to his work with lions and other large predators - when he first started working at Lion Park, there were things he was taught about interacting with lions that he's ultimately disregarded.
Many keepers/handlers will carry a stick when they go into a lion enclosure. They won't look the cats in the eye, they won't raise their voices, and they never get down on the ground or turn their back on them. The Wikipedia article about Richardson that I linked above says he's dispelled many myths about the way to care for and handle big cats like lions - I don't know how true that is, because Richardson continuously emphasizes the fact that his approach to the lions seems to work for him, but he in no way advocates for a radical change in the way most people would interact with them.
Richardson knows the lions that he works with, and he has different relationships with all of them. Some lions he considers "acquaintance lions," lions he says hello to and they abide in mutual tolerance of each other. Some lions, like Tau and Napoleon, or Meg and Ami, who he knew since they were young cubs, he considers to be his brothers and sisters.
And he says that just as with humans, the nature of his relationships with individual lions sometimes changes. He notes that in his experience, a lion is at its most dangerous when it's in its "teenage" years, about two to three years old. A lion in its teens, he says, is big enough to kill you, and young enough to want to. He has not gone through his experiences with these cats unscathed - the book opens on Richardson recounting an incident that occurred when he entered the enclosure of a young lion named Tsavo. He was showing his family around Lion Park for his nephew's birthday, and after playing football with Tau and Napoleon he wanted to show Tsavo to them, who was a little bit older.
Tsavo is not one of those lions that Richardson knew from when he was a cub. Tsavo was mistreated at another facility and moved to Lion Park during his crucial teenage lionhood period. Richardson is very clear about the fact that he ignored the signals that Tsavo was giving him not to approach, but he ignored those signals because he knew his family was watching. Unfortunately for his family, they got to see Richardson take a massive paw to the face (which might have killed him if Tsavo had not been de-clawed at his first home) and subsequently be straddled by an angry, biting lion.
Richardson was able to get away with minor injuries when another handler intervened and distracted Tsavo. He learned that even animals with whom he has a good relationship will react differently when there are strangers present, and that just like humans, animals will have good days and bad days. And when you weigh 500 pounds and are packing pawfuls of claws and mouthfuls of teeth, you get to dictate the terms of any given interaction.
I think that Richardson is successful in his animal relationships because he relates to them as individuals, and does his best to remain in 'animal headspace' while interacting with them. I'm not sure if he occasionally falls into the trap of anthropomorphizing animals or if he's just colorful in the way that he talks about them as individuals, but I think for the most part he avoids projecting human thoughts and/or feelings on them in the moment.
Though Richardson's book is subtitled My Life Among the Big Cats of Africa, most of it is devoted to his beloved lions, and secondary emphasis is placed on hyenas. He relates a few anecdotes about a South American jaguar, a few cheetahs, and leopards, but because Richardson is a behaviorist it's mostly about the specific interaction. Very little about the ecology of these animals' wild counterparts is included in the book - as I said, Richardson is very "in the moment" as opposed to coming off like an academic. This is unfortunate, because he mentions that there are other types of cats at the facility, like caracals, and we hear nothing about them.
If you're looking for a book about zoology this will disappoint you. If colorful memoirs from non-professional writers appeal to you, you'll likely quite enjoy it.