Circular Reasoning
Logical fallacyAssuming the conclusion somewhere among the premises, so the argument never supplies independent support. The formal name is begging the question, or petitio principii.
An everyday example
“The rule is fair because it treats everybody exactly as they ought to be treated.”
The same move in this debate
“Animals are food animals, which is why it's acceptable to eat them.”
Where this appears
That animals exist for us to eat is precisely the point in dispute. Offering it as the reason for eating them assumes the conclusion rather than supporting it.
Describing the choice as personal presumes that no one else's interests are at stake, which is exactly what is in dispute. The description does the work the argument was supposed to do.
On the moral reading, asserting that animals have no rights is a restatement of the conclusion rather than a reason for it.
Asked why only humans count, the speaker answered 'because we are human'. That uses the conclusion, that human membership is what confers moral standing, as its own support. When pressed for something more, he listed consciousness and the capacity for pain, which is a different argument, and one animals also meet.