Only humans count, because we are human
'Humans matter because they are human' is a restatement, not a reason. When people explain why humans matter, they name things like feeling pain, having a life of one's own and caring what happens next. Many animals share those. Favouring your own species may be natural, but it does not show that other animals count for nothing.
Being human is not an arbitrary label. It marks membership in a kind whose members reason, make moral choices, hold one another to account and form communities of mutual obligation. It is reasonable to give that community a special standing, just as we owe more to family than to strangers. Animals can't take part in morality the way people do, so it isn't wrong to put humans first and treat animals as outside that circle.
Being human can't be the whole reason, because the sentence just repeats itself. Ask why being human matters and the answer is either 'because it's human', which goes in a circle, or a list of features. In the exchange this came from, the list arrived straight away: being conscious, feeling pain and pleasure, having a life to live. Those features carry the moral weight, not the species label.
Those features are not unique to humans. Pigs, cows, chickens and fish feel pain, fear and distress. They prefer some states to others and act to avoid harm. Humans have capacities other animals lack, such as language and long-range plans, and those differences are real. They can justify treating humans differently in many ways. They cannot justify treating an animal's suffering as worth nothing. Whether something can be hurt does not depend on whether it can do algebra.
The argument from kind does not rescue it. Some say that humans as a kind are rational and moral agents, so every member counts, including newborns. That explains why we protect infants, but it is not a reason to ignore everyone outside the kind. A being's pain is bad for that being, whatever group it belongs to.
Partiality is a fair point, and most people accept some of it. You may owe more to your family than to strangers, and more to humans than to animals. But owing more to some does not mean you may harm others when you don't need to. Caring more about your own children does not entitle you to hurt a neighbour's child for a better meal. The question about animals is not whether they rank equal to humans. It is whether their suffering can be disregarded when the reason is taste. 'We are human' doesn't answer that question.
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: 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.