In chapter two, the pigs announce that they have taught themselves to read and write:
The pigs now reveal that during the past three months they had taught themselves to read and write from an old spelling book which had belonged to Mr. Jones's children and which had been thrown on the rubbish heap.
The other animals are impressed with the pigs' education. Not only have the pigs taught themselves to read and write, but they have created the principles of Animalism. The other animals like the sound of Animalism. This is one reason the other animals accept the pigs as leaders. The pigs' philosophy sounds good. It makes the other animals feel there is hope. The other animals want to see change and the pigs are now offering hope. The pigs have named the farm Animal Farm, and the other animals like the sound of Animal Farm.
Also, Snowball is quite the leader and is later honored with a medal for bravery:
Snowball shows his expert use of military strategy during the attack—which becomes known as the Battle of the Cowshed—and is later awarded a medal.
No doubt, the other animals recognize the great leader that Snowball is. He is organized in his efforts. The other animals accept his expertise and consider his ideas effective.
Also, it was Old Major, a pig, who initially planted the desire to rebel against Mr. Jones and his animal cruelty:
The animals gather in the barn as Old Major, the prize boar, tells them that he has thought about the brutal lives that the farm animals lead under human bondage and is convinced that a rebellion must come soon, in which the animals throw off the tyranny of their human oppressors and come to live in perfect freedom and equality.
The animals respected Old Major and trusted him. It makes sense that the pigs would take up leadership where Old Major left off. After all, everything Old Major talked about made sense to the animals, or at least most of the animals. What animal would not desire perfect freedom?
Most importantly, the pigs have created a commandment that all animals are equal. This is important to the other animals and they accept the pigs as their leaders for they talk a good talk.