The farm changing to a republic symbolizes the ascent of Napoleon as supreme leader of Animal Farm: in fact, it symbolizes the turning point when the farm is about to become a tyranny. 

Earlier, the animals had been taught to anticipate a republic as a good thing, a sign of animal equality and solidarity:

The flag was green, Snowball explained, to represent the green fields of England, while the hoof and horn signified the future Republic of the Animals which would arise when the human race had been finally overthrown.

Then, the April after Snowball is run off the farm

Animal Farm was proclaimed a Republic, and it became necessary to elect a President. There was only one candidate, Napoleon, who was elected unanimously.

The declaration of the "Republic" and the election of Napoleon as President are both merely ruses to allow him to seize power. Now that he is in charge, he will run roughshod over the rights of the other animals. This move to a "Republic" also represents or allegorizes the betrayal of the communist revolution in the USSR by Stalin, who similarly took control and became a tyrant.