First of all, they are very cute. Secondly, when you feed your pets, they give you collection items, which in Wild West is a very lucrative affair. If you manage to assemble a full pet collection, you can trade it in for a valuable gift through the collection book. You will find the book next to your house on the farm.
Whenever you complete a collection, an exclamation point will appear above the book. This means that it is time to exchange your collection for a reward. Each pet has its own collection. The more often you trade a particular collection in, the larger the gift that you get in exchange will be.