Napoleon takes the puppies away from Jessie and Bluebell as soon as they are weaned because he wants to use them as a private security force.
Napoleon tells the mothers that his taking them is an advantage.
As soon as they were weaned, Napoleon took them away from their mothers, saying that he would make himself responsible for their education. (Ch 3)
When Napoleon takes no interest in Snowball’s committees, and says “the education of the young was more important than anything that could be done for those who were already grown up” (ch 3), he is really cementing himself in a position of power.
This early on, Napoleon is already takes steps to make sure he and not Snowball is in charge. When it is time to drive Snowball out, the puppies re-appear as “nine enormous dogs” and run straight for him (ch 5). They are “as fierce-looking as wolves” and keep close to Napoleon. He later uses them to tear out the throats of animals he says are traitors.
Compared to other animals on the farms, dogs have an advantage. They are strong, intelligent, and have very sharp teeth. It is for this reason that Napoleon takes the puppies as soon as they are born. He keeps them away from their mothers until they are adults, when he brings them out as enforcers. Other animals are less likely to question Napoleon and the pigs when they are backed up bysnarling dogs.
The taking of the puppies represents the importance of a security force loyal to the totalitarian leader. Napoleon uses the dogs to keep the other animals in line.