The argument from marginal cases claims that you can't both think that humans matter morally and that animals don't, because no reasonable set of criteria for moral worth cleanly separates all humans from all animals. For example, perhaps someone says that suffering only matters when it happens to something that has some bundle of capabilities like linguistic ability, compassion, and/or abstract reasoning. If livestock don't have these capabilities, however, then some people such as very young children probably don't either.
This is a strong argument, and it avoids the noncentral fallacy. Any set of qualities you value are going to vary over people and animals, and if you make a continuum there's not going to be a place you can draw a line that will fall above all animals and below all people. So why do I treat humans as the only entities that count morally?
If you asked me how many chickens I would be willing to kill to save your life, the answer is effectively "all of them".  This pins down two points on the continuum that I'm clear on: you and chickens. While I'm uncertain where along there things start getting up to significant levels, I think it's probably somewhere that includes no or almost no animals but nearly all humans. Making this distinction among humans, however, would be incredibly socially destructive, especially given how unsure I am about where the line should go, and so I think we end up with a much better society if we treat all humans as morally equal. This means I end up saying things like "value all humans equally; don't value animals" when that's not my real distinction, just the closest schelling point.
 Chicken extinction would make life worse for many other people, so I wouldn't actually do that, but not because of the effect on the chickens.
I also posted this on my blog.