Personally, I am somewhat sympathetic to this position. I don't think it follows from the fact that animals are sentient that animals can reason. Certainly they don't have conceptual thought and language, not even remotely the way humans do. And if they did posses full rights, it wouldn't follow that you shouldn't just kill them, but you couldn't compel them at all. I want to clear out some land and build a farm, well I can't because some muskrat of some sort has made it his home. Silliness follows from this.
But I am somewhat sympathetic, like I said, to some sort of basic animal rights, to be differentiated from human rights. It is clear that they are sentient beings, and have some sort of basic level of awareness and free will, they have emotions and personality, research shows even that some of the more intelligent ones can abstract and even form some first level concepts. I think this leads to a certain very basic level of protection, that you can be compelled by the law not to cause unnecessary suffering and cruelty to animals. They can still be killed and eaten, can still be used for our ends and purposes, but that has to be done within certain cruelty laws.
Idon't think most objectivists believe this, unfortunately, they believe it to be monstrous and immoral, of course, just that the law cannot address it.
Interestingly enough, it seems that Ayn Rand was also very sympathetic to the idea of animal rights, she just thought ultimately it couldn't be proven, according to Barbara Branden in an interview in Liberty Magazine.
