When the Constitution was written, the idea of political parties was not in it. It was felt at the time that the political process could get by without parties.
But very early on, a rift happened. Alexander Hamilton (and to a lesser extent John Adams) felt very strongly that we should have a central bank, and incur a national debt so we could develop credit. They felt that the central bank should incur the debt of the states, and in turn borrow from foreign governments to centralize government and stimulate the economy. This group was known as the “Federalists”. George Washington was a Federalist sympathizer although he never really came out as a Federalist.
This Federalist idea was strongly and bitterly opposed by Thomas Jefferson and the Southern coalition. By the time George Washington had finished his two terms in office, This opposition coalesced into a party, called the Democratic-Republicans.
When John Adams ran for President he was a Federalist, and Jefferson ran against him as a Dem-Rep. Adams won, and Jefferson became VEEP per the constitution at the time.
The final straw happened during Adams’ presidency when he signed the Alien and Sedition Acts, which allowed the government to jail people who were critical of the government. Jefferson bitterly opposed the acts (which would certainly be declared unconstitutional today) and in 1800 won the presidency in large part on the platform of repealing the acts. (Ironically, Jeffereson actually used the Sedition Act to jail a few opponents of him).
So the multiple party system was arguably an inevitable consequence of how democracy is organized. In an English-style parlamentary (sp?) system you can have multiple parties forming coalitions. In our constitutional republic style, the two party system was probably inevitable based upon the geography of the rural South and the urban North.