Animals have no interest in continued existence, so cannot have a right to life
Applied consistently, this criterion also excludes human infants and people with severe cognitive impairment. Either it is applied to them too, or an exception is being made without a stated reason.
Death harms a being by depriving it of the future it wants and expects, and that requires being able to see oneself as a continuing self with plans, hopes and a life story still to unfold. A cow or a chicken lives in the present: it can suffer, and causing that suffering is wrong, but a painless death takes nothing from it that it was ever aware of having, because it has no conception of a future self to lose. A right to life protects an interest in continuing to exist, and a being that cannot hold that interest cannot be wronged by losing a life it never projected forward.
The argument holds that a right to life requires the capacity to conceive of oneself as persisting through time with a future worth having. On this view most animals, lacking that self-conception, cannot be harmed by death as such: only by pain preceding it.
The difficulty is the argument from marginal cases. The same criterion excludes human infants, people with advanced dementia, and the severely cognitively impaired, all of whom we take to have full moral status. Any threshold set high enough to exclude animals will catch some humans; any threshold low enough to include all humans will catch many animals. Philosophers including Singer and DeGrazia have argued that the defensible threshold for basic protection is sentience rather than future-directed self-awareness. There are serious replies to this (some rely on species-typical capacities rather than individual ones) but the objection as usually stated does not engage them.
Applying a principle generally while claiming an exemption for one's own case, without giving a relevant reason for the exception.
How it applies here: The criterion is applied so as to exclude animals, while the humans it would also exclude are quietly exempted. No reason is given for why the standard binds in one case and not the other.