If animals have rights, doesn’t that mean we have to intervene to stop animals from killing other animals, or that we must otherwise act affirmatively to prevent harm from coming to animals from any source?

No. the basic right not to be treated as a thing means that we cannot treat animals exclusively as means to human ends–just as we cannot treat other humans exclusively as means to the ends of other humans. Even though we have laws that prevent people from owning other humans, or using them as unconsenting biomedical subjects, we generally do not require that humans prevent harm to other humans in all situations. No law requires that Jane prevent Simon from inflicting harm on John, as long as Jane and Simon are not conspirators in a crime against John or otherwise acting in concert, and as long as Jane has no relationship with John that would give rise to such an obligation.

Moreover, in the United States at least, the law generally imposes on humans no “duty to aid” even when other humans are involved. If I am walking down the street and see a person lying passed out, face down in a small puddle of water and drowning, the law imposes no obligation on me to assist that person even if all I need to do is roll her over, something I can do without risk or serious inconvenience to myself.

The point is that the basic right of humans not to be treated as things does not guarantee that humans will aid other humans, or that we are obligated to intervene to prevent harm from coming to humans from animals or from other humans. Similarly, the basic right of animals not to be treated as things means that we cannot treat animals as our resources. It does not necessarily mean that we have moral or legal obligations to render them aid or to intervene to prevent harm from coming to them.