Animal Rights and Stupidity

Look at this site. I was shocked to come across this sentence: "We do not believe that animals have rights but we do believe that people have responsibilities." Question: Who are the beneficiaries of these "responsibilities"? There are two possibilities: humans and the animals themselves. I assume this organization is not making the Kantian claim that our responsibilities to animals are really just duties to humans who happen to be interested in them. So the beneficiaries of our responsibilities are the animals themselves. But how does this differ from saying that the animals have rights? If our responsibilities are to refrain from harming animals, then they have negative rights against us. If our responsibilities are to promote the welfare of animals, then they have positive rights against us. I'm starting to think that people who deny animal rights are stupid. See here.