There are problems with the traditional liberal basis of human, and other, rights, as embodied for instance in the UN declaration:
The liberal view of rights is based on the individual, and focuses on what is due to the individual. It sees the individual as a fundamentally distinct entity, and cannot take into account in any sound way the environment and context of the individual.
A fundamental Green concept, though, is interconnectedness, relatedness. We are not fundamentally isolated, independent entities but are entities within a web of relationships. And these relationships are not just contingent but are fundamental.
It is these relationships that provide the ground for what is due to each entity - whether it be human, animal or anything else. As a result, the focus shifts from rights to responsibilities. And to talk of somebody's 'rights' is merely shorthand for talking about the responsibilities that we all have to them.
One major benefit of this approach, based on relatedness, is that it treats animal rights in the same way as human rights, but without assuming that those rights are identical. What is due to an animal is determined by the relationships that are appropriate to the animal, and what is due to a human is determined by the relationships that are appropriate to the human. (I find this happens to link nicely with the juridical aspect in Dooyeweerd's system of thinking.)