Question

From the anthropocentric perspective, the idea that we should curtail human activities to preserve a nonhuman species is generally anchored in which of our ethical theories?

Answer 1

This theory would be "utilitarianism" - the belief that we should maximize the "utility" - the well being of everyone. 

So this idea believes that animals should not die for our entertainment because our benefit (say, the joy of hunting) is less than their suffering (at being shot by a hunter).

It is opposed to an anthropocentric view, which says that we can use animals for whatever purpose we see fit.