The surface meaning of the subtitle is that the story is styled as a fantasy; animals talk and perform human work, and are eventually recognized by society as intelligent. Even the initial rebellion is only remarked on for its audacity, not for the talking, thinking animals. In this way, the story is a fable, a moralistic tale like many "fairy stories" from older times. With a happier tone, it could even be a children's story.
The other main reason is that the book was published during a contentious time in history; Russia had undergone its own revolution and publishers were wary of political books, not wanting to incite anger. Adding the subtitle allowed the book an extra layer of fiction, despite the blatant allegories inside. The book could therefore be dismissed by potentially dangerous critics as a "fairy story" instead of a biting satire, while readers could judge the qualities of the work for themselves.