Because Animal Farm is an allegory for the rise of Communism, Animalism is intended as a broad satire of that philosophy, showing its weaknesses and failure to bring about substantial change. As the novel progresses, Napoleon the pig changes the rules to favor himself and oppress the other animals, just as in real life Communist leaders changed their principles to give themselves an advantage over others.
These three had elaborated old Major's teachings into a complete system of thought, to which they gave the name of Animalism. Several nights a week, after Mr. Jones was asleep, they held secret meetings in the barn and expounded the principles of Animalism to the others.
(Orwell, Animal Farm, george-orwell.com)
The founding principle of Animalism is "All animals are equal." When all animals have an equal share in the outcome, and no one animal is favored over another, the system seems to work. However, as the rules change, the principle becomes "All animals are equal, but some animals are more equal than others." In real life, Communism usually ends with some people in positions of absolute power, controlling and leeching from the hard work of the people underneath them, while pretending they are sacrificing "for the good of the Collective." This often ends in new revolution as the people start to resent their oppression.