This is a good question. From the beginning of the book, the pigs are the smartest of the animals. We can see this in Old Major and later on in Napoleon, Snowball and Squealer.
The pigs are so intelligent that they learn how to read and write. So, they run the farm and the principles behind Animalism. The animals come to learn these rules in three ways.
First, Napoleon speaks them to the animal and reasons with them. For example, he summarized the heart of Animalism to the rest by saying "four legs good, two legs bad.
After much thought Snowball declared that the Seven Commandments could in effect be reduced to a single maxim, namely: ‘Four legs good, two legs bad.’ This, he said, contained the essential principle of Animalism.
Second, he writes the commands down, so that all the animals can see what the laws are. Moreover, he makes a point that these laws were the fruit of much labor and thought.
They explained that by their studies of the past three months the pigs had succeeded in reducing the principles of Animalism to Seven Commandments. These Seven Commandments would now be inscribed on the wall; they would form an unalterable law by which all the animals on Animal Farm must live for ever after.
Third, he uses tons of rhetoric through Squealer, a golden tongued pig. We can say that Squealer is a propagandist. Here is an example of Squealer's powerful rhetoric:
Comrades!’ he cried. ‘You do not imagine, I hope, that we pigs are doing this in a spirit of selfishness and privilege? Many of us actually dislike milk and apples. I dislike them myself. Our sole object in taking these things is to preserve our health. Milk and apples (this has been proved by Science, comrades) contain substances absolutely necessary to the well-being of a pig. We pigs are brainworkers. The whole management and organisation of this farm depend on us. Day and night we are watching over your welfare. It is for your sake that we drink that milk and eat those apples. Do you know what would happen if we pigs failed in our duty? Jones would come back! Yes, Jones would come back! Surely, comrades,’ cried Squealer almost pleadingly, skipping from side to side and whisking his tail, ‘surely there is no one among you who wants to see Jones come back?’
Squealer is a pig who works closely with Napoleon. The animals on the farm usually hear orders out of Squealer's mouth. Whenever Squealer tells the animals what Napoleon desires, he always says "Comrade Napoleon said so". There are pigeons that also act as messengers in the story, except the pigeons spread news of rebellion outside of the farm, not within Animal (Manor) Farm.