I believe the rebellion started with the speech from Old Major who encouraged the animals to look at how they were being treated and realize there was a better way where all animals could be equal.
When Old Major died, he then became a hero of sorts, and his message became legend among the animals who only talked about it until the night that Farmer Jones and his men forgot to feed them for a long period of time.
Combined with the other mistreatment, the starvation finally got the animals mad enough to kick in the feed stall so they could eat. When Farmer Jones tried to get them back under control, they had already gotten a taste of freedom and they wanted to keep it so they fought back.
After that, the pigs took over and Snowball studied the battle plans of Cesar so he would know how to fend off further attacks. It worked, as the animals were able to turn back the humans at the battle of Cowshed to finally take control of their farm.
Once in power, the animals changed the name of the farm from Manor Farm to Animal Farm, flew their own flag, let the pigs take over as leaders, and came up with the motto, "All animals are equal."