The problem isn't political or historical quite as much as structural.
Pretty much everywhere except a few city governments, the US uses what's called a first-past-the-post electoral system. In other words, whoever has the most votes (a plurality) wins outright. Since practically all representative districts in the US (again, outside of city governments) are single-member that means any plurality in votes is rewarded with universal representation.
In this political situation, to win you want as many people voting for the same candidate as possible (obviously). But to counteract that, any opposition ideally should settle on a single candidate as well if they have any hope of winning. The only way to integrate such an electoral process into a political party system is to make two large "big tent" parties and absorb as many possible coalitions and positions into one party or the other. Thus, you get the two-party system, be it Federalists vs. Democrats, Whig vs. Democrats, Democrats vs. Republicans.
The problem is further compounded by the way the President is elected. The electoral college isn't explicitly ordained to be structured this way, but taking a plurality of the votes in a state gives you their entire electoral vote. So, if one party in California take 49% of the vote to the other party's 48%, the first party takes 100% of its electoral vote, in this case 55 votes of a total of 538. That only further encourages the two-party strategum outlined above.
For a group of people who had hoped to avoid the formation of political parties, especially any two-party system, they couldn't have conditions less favorable to a two-party stranglehold. And even so, the US has sustained more than two parties at multiple points in its history. The Democratic Party especially was divided between the Democratic party proper of the North and the more homogeneous caucus of Southern Democrats, which can often be referred to as separate party. Yet even here, during all but one election (that of 1860), the Northern and Southern Democratic parties settled on one candidate - though typically with much overwrought compromising and deal-brokering. To accomplish this, Democratic National Conventions almost always honored a rule mandating that a candidate receive not a majority, but a 2/3 supermajority of approval from delegates. The early-middle 1800s supported the Whig Party, Democratic Party, and the American Party (Know-Nothing Party) simultaneously, though not especially comfortably. Other influential third parties whose platforms were adopted by one or the other major party (eventually and only in part) are the Free Soil Party, various iterations of Progressive Party, the Greenback Party, the Libertarian Party, and even the Communist Party.
If you want to see more than two parties sustainably supported by American politics, you need to change the way elections are performed more than anything else. Make it either a Party Proportional system (less likely) or some version of a Ranked Voting system (more likely).
If you want to see what those parties might look like, just look at the current caucuses that make up the Democratic and Republican parties and then throw in the current larger "third" parties (Greens and Libertarians chiefly).