I know that at the time the Constitution was drafted partisan political parties did not exist, not in the way that we conceive of them today, and George Washington did not have a political party. However, John Adams was a member of the Federalist Party, and the Federalists were the first political party in the US. What factors led to other political parties taking shape in the US?
The Federalists and Anti-Federalists started around the Constitutional Ratification, during the adoption fight but eventually grew under Hamilton to the political party that they became during the first few presidential administrations. Afterwards you had like-minded groups grow because the only way to get elected, or names on the ballots because of the way the political system was structured was through parties. Remember initially there was no direct vote, you elected your State Representatives and Senators were elected by State Legislatures so to get a sizable block you needed to get a sizable number of like-minded people into State government and the House of Representatives.
Since that time there have been the Democratic and Republican parties that we have come to know today, as well as others that rose out of the times. Others rose and feel depending on feelings on political power, such as Andrew Jackson and what became the Democrats where he favored the Executive branch while Henry Clay and the Whigs favored the Legislative. This was a time of political friction, and it is during times like these when partisan politics become intense (like the past few years in the US with the rise of the Tea Party) when new groups form.
A couple of short overviews can be found here:
Both the US and British legislative bodies underwent transformations in the last half of the 18th and first half of the 19th century, and it is useful to study them together. At the beginning of the period, neither had "political parties" as we understand them. They evolved political parties as a way of forcing a diversity of viewpoints into actionable legislation, of dealing with issues, and most importantly of securing and maintaining a hold on power long enough to carry out a legislative program.
Each of the other answers has supplied the core facts; I quibble with the emphasis given in various places, but I agree with the other respondents. The fascinating aspect of the question is that both the US and British systems, operating independently, evolved similar structures despite the fact that both systems claimed to vehemently oppose those structures.
A couple of other notes. First, the founding fathers not only opposed parties, they opposed campaigns. Candidates were supposed to be "disinterested"; their supporters could advocate on their behalf, but public service was an obligation suffered honorably, not something to be sought. That meant that political activity in the sense of modern political parties was shameful and effectively disqualified the candidate from office. Obviously that ethical position was eroded and destroyed by the forces that create political parties.
I think the second fascinating trend was appointments/patronage/placement. One of the primary reasons for the revolution was the British system of "placemen" that failed to either effectively include/coopt Americans or exclude/disenfranchise Americans (depending on your point of view). The first couple of administrations tried strenuously to avoid creating this system. One of the Presidents (I believe it was John Quincy Adams, but I will welcome corrections from H:SE) inherited an administration full of his political opponents, but refused to dismiss them and appoint replacements who would work with him. He did not want the reputation of someone who would appoint his political friends. Again, this ethical position was eroded away by the forces of political parties. (One of my revelations while watching the movie Lincoln was the direct line between British placemen and the "shady dealings" of Mr. Bilbo and his ilk.)
I haven't studied it in depth, but the dominant individual seems to have been Martin Van Buren, who created the first recognizeable political machine. He was backed by a political party and allocated patronage jobs to members of the party. Note that there was a key difference; the British placemen received their positions in return for loyalty to the crown; Van Buren's political machine offered patronage in exchange for loyalty to the party. That meant that anyone could elect to be a member of the party, work for the party's benefit, and receive patronage. That's a big part of the power and the justification of the American political machine.
I've cited Gordon Wood's Radicalism of the American Revolution; the last third of that book has a fairly decent (albeit unfocused) history of the rise of the American political party. Reaching back into my memory I believe my other favorite source on this topic was "Adams vs Jefferson: the tumultous election of 1800".
Jefferson and Adams' policy debates over Revolutionary War debts and relations with France and Great Britain became very public and personal throughout Washington's presidency and into the Presidential election of 1796. Their surrogates circulated vicious personal attacks during Adams' Presidency, while Jefferson served as Vice President. Their followings became more and more divided as precedent was being established for America's stance on issues including naval, fiscal, and diplomatic policy. The conflict came to a head during the debates of the Alien and Sedition Acts. The the election of 1800 was decided by congress, which Aaron Burr and Alexander Hamilton manipulated, and sharply defined party lines, creating a victory for Vice President Jefferson over President Adams. The 12th Amendment regarding the election of Vice Presidents then precipitated the need for more organized national campaigns.
What led to the rise to political parties was the fact that Hamilton and Jefferson had conflicting views. Also the fact that Washington favored Hamilton’s ideas which made Jefferson very mad because he wanted to have his ideas favored also. Last but not least the two parties fought over governmental issues also played a part in the rise of political parties.
It boils down to simple mathematics. The US electoral system is mostly based on a winner-takes-all approach (BTW, that's mostly not written into the Constitution, but rather evolved ad-hoc, for similar mathematical reasons).
In a winner-takes-all system, only the two biggest vote-getters will ever have meaningful influence, so it is natural for a dualism to evolve.
The other factor is that in politics, nothing can be accomplished by yourself, only by teaming up into some form of coalition and alliances. For the most part, long term coalitions and alliances are far more valuable to all participants (because they allow calling in favors far in the future). Take that together with the natural dualism inherent in the electoral system, and you have the making for a very stable two-party system.
Interestingly, the same mechanism is also driving another aspect: diversity within each party. Many European democracies don't use a winner-takes-all system, and as a result have more parties participate in the democratic process. The price for it, though, is that parties are far more homogeneous internally than US parties, to the point that party leaders can enforce party-line votes by controlling who will even be on the ballots.
Political parties exist in every country that has elections. America has elections, and there is nothing special about America.
The question that you should ask is why does a representative system always lead to political parties? A few dictatorships don't have them, but there is no election-based country that has no parties, even if the country is not fully democratic.