The answer to this can be found in Chapter 9 of the book. There, Diamond is trying to explain why so many places in the world did not have large domesticated animals. This is important to his theory because some people would claim that people like the Australian Aborigines did not domesticate animals because they were too backwards. Diamond disagrees with that sort of assessment and wants to say that geographic luck led to most of the inequality in human history.
The term “Anna Karenina principle” refers to the book Anna Karenina, by Leo Tolstoy. The first sentence of that book is “Happy families are all alike; every unhappy family is different in its own way.” This relates to animals because it means that there are many things that must be true about an animal in order for it to be undomesticable. That means that there are many different ways that an animal can be “bad” for domestication, but all domesticable animals are similar because they have all of the necessary traits.
Diamond says (beginning on p. 169 in the paperback edition of the book) that animals must have the right diet, growth rate, ability to breed in captivity, temperament, calmness, and social structure in order to be domesticated. An animal can have all but one of these and be useless for domestication. Therefore, “unhappy” animals can be unhappy in different ways but all “happy” (domesticable) animals have to be very similar.
This supports Diamond’s theory because it means that the ability to domesticate animals comes about by luck. You have to be lucky enough to have one or more of the few animals that fulfills all the criteria living near you in order to domesticate them. People like the Aborigines did not have that luck.