When interpreting Animal Farm, it’s important to recognize that the story is highly allegorical. This means that the dogs, rats, pigs, and other animals and people in the tale represent something else besides animals.
On the surface, Animal Farm is about farm animals who decide to revolt against their human masters. In Chapter 1, Old Major, a pig discussing a future rebellion against the humans, is giving a speech about how farm animals don’t deserve to be enslaved and killed by humans. During this speech, some rats come forward to listen, and the dogs see them and chase them.
When Old Major sees this, he asks the crowd whether rats count as animals, implying that rats and dogs are on the same side against humans.
Here, the book is using the animals to represent different types of people. The humans represent those in power in society. The animals represent those who aren’t.
Dogs are owned by people, usually stay near the human dwellings (unlike the other farm animals), and humans tend to value dogs more than other animals. So, given all of this, we can surmise why dogs chase rats. After all, humans don’t like rats in the house.
Animal Farm is partly about how we decide to choose sides in conflicts.