The pigs simply take the milk. The rationalization they use is that they need the milk in order to be intelligent. They threaten the animals with the possibility that the humans may return and the pigs need to be as smart as the human in order to run the farm and protect it from all enemies. The rest of the animals agree to this because they don't want to see the humans return. The suggestion is that the pigs are already plotting to take the place of the humans in the hierarchy of the farm. Jones took all the milk, so they want all the milk. The next time the pigs want something, it will be easier for them to take it.
The pigs gain rights to the milk simply by saying that it is their right. The important thing that this suggests is that, althought the "personages" have changed, the rules of power still remain in play ... it abhors a vacuum. It also suggests that most "people" are in fact sheep and are easily susceptible to propaganda. What ultimately is suggeted is that the state of the animals on the farm will, in the long run, not be much different from their initial state.
The pigs gain the rights to the cow's milk by simply convincing the other animlas that they need the milk to stay healthy so that they can continue to manage the farm. The animals allow this because they believe that the pigs have the ability to keep them safe from Mr. Jones and other humans like him. This event suggests that the power hierarchy on the farm is one that is based on the premise that the pigs are the "leaders" of the farm. The other animals almost innately look to them for the answers. Ironically, the animals are taking the place of the humans that ran the farm prior to the rebellion.