Animals have the right to equal consideration of their interests. For instance, a dog most certainly has an interest in not having pain inflicted on him or her unnecessarily. We therefore are obliged to take that interest into consideration and respect that right.
However, animals don't always have the same rights as humans because their interests are not always the same as ours and because some rights would be irrelevant to animals' lives. For instance, a dog doesn't have an interest in voting and therefore doesn't have the right to vote, since that right would be as meaningless to a dog as it is to a child.