The development of the two political parties during this time can be explained by two things.
First, there were divergent points of view on domestic politics. Factions formed, lining up behind Thomas Jefferson and Alexander Hamilton, respectively. Jefferson wanted more democracy and an egalitarian society made up largely of small farmers. Hamilton wanted less democracy and an industrial society in which inequality was accepted.
Second, there were the beginnings of conflict on foreign policy. The war between Britain and France led to disputes between the Jeffersonians and Hamiltonians. The Jeffersonians were in favor of France while the Hamiltonians favored Britain.
Because of these sorts of differences in political views, the two party system developed.