What is the motif of the animals throughout the novel, and how is it being used throughout the story? What theory is the author intending to show with this motif?

The central motif of We The Animals is the depiction of children as animals. A secondary motif broadens the original one to include all humans in the story. In this particular "animal farm," some animals are also more equal than others—or, at the very least, they think they are. Adult onlookers describe the children as "dogs" and "locusts," but they are really no better. In fact, if anything, the kids are more human than the adults, filled as they are with normal human foibles, desires, and insecurities.
However, the point of view is important here. We are given a child's view of the world, one in which it is the adults who seem like strange, exotic creatures:

Her mascara was all smudged and her hair was stiff and thick, curling black around her face and matted down in the back. She looked like a raccoon caught digging in the trash: surprised, dangerous.

The brothers are practically feral children. They frequently live down to the low expectations set for them by society. They fight, they get into scrapes, and they flip off their neighbors. However, at the same time, they are trying—in their own imperfect way—to find a place in the world and to forge their own identities without parental guidance in an environment hostile to mixed-race relationships. By getting in touch with their raw animality, the boys are also simultaneously discovering what it means to be human. These kids may be wild, but they are still kids. Unlike the adults in the story, this particular species of animal evolves.
The narrator's developing sexuality is also related to the animal motif. Over time, he comes to realize that he is gay. Along with his mixed-race heritage, this adds to his outsider status. In a harsh, brutally unforgiving environment, he is now doubly "other," not just estranged from the world around him, but also from his brothers. Their violent homophobia confirms that there is a distinct hierarchy in this "animal kingdom." If you are gay, like the narrator, then you are not just inhuman, you occupy the lowest ladder on the evolutionary scale.
The animal motif allows Torres to get to the truth with pinpoint accuracy. He does not preach; he does not judge; he is not making grand, sweeping statements about the human condition. What he does do, however, is describe what happens with unerring detail. This is what we would expect if we were reading an animal fable. The harshness of nature needs to be laid bare by keen observation of what unfolds before our eyes. In We The Animals, that is precisely what Torres does.