Animals are here for us to eat
This states a conclusion rather than supporting one. Nothing about an animal's existence confers a purpose, and the populations in question exist because we bred them.
Farm animals did not wander into our lives by chance: over thousands of years humans bred cattle, pigs, and chickens into being for food, and those animals exist in the numbers and forms they do only because of that arrangement. Eating them is not a betrayal of what they are but the fulfilment of the role they were shaped for, part of an order in which humans sit above other animals and have always lived by them. Without that purpose most of these animals would never have been born at all.
The claim assigns a purpose to a category of beings and then treats that purpose as a justification. But purposes of this kind are attributed, not discovered. A pig has interests of its own (it avoids pain, forms attachments, and acts to continue living) and none of them concern us.
The circularity becomes visible when the reasoning is generalised: any group could be said to exist for a given use once we have arranged for them to be produced for it. We created these populations through selective breeding over centuries, which explains why they are here but does not establish what we may do to them.
Assuming the conclusion somewhere among the premises, so the argument never supplies independent support. The formal name is begging the question, or petitio principii.
How it applies here: 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.
Moving from a description of how things are to a claim about how they ought to be, without supplying the moral premise that would connect the two. Identified by David Hume.
How it applies here: Moves from the fact that we do breed animals for food to the claim that this is their purpose, with no premise connecting the practice to the entitlement.