It is important for you, the caregiver to let your animal know that they are greatly loved and have contributed greatly to your life and that you will miss them. However, since animals are so stoic, they will do their best to hang on just to please you. It might be a good idea for you to let them know that they are to think of themselves first here and that it is OK for them to make their transition. Remember that animals, especially dogs are here for us. They try to make us happy.
Cats are a bit more individual, but they also are concerned about leaving.
However, animals view death differently than we do. I think they take it more matter of fact. Although they do and will grieve. And they grieve in their own way, especially the ones that are left behind with you. Just as with humans, our pets have shared our lives and there are many occasions that stick out in our memories of them.