There's nothing in the Constitution about political parties, two or otherwise. In fact, the initial idea at the time was that the US wasn't going to have political parties and that voters would focus on individuals instead.
Whether or not the Constitution says anything about political parties, it establishes a "winner takes it all" voting system that has the effect described above. The Republican party should long ago have split into a loony Tea Party fraction and a conservative fraction, and half of the Democrats would politically be more at home in the Green party.
But these things are not realistic with a first-past-the-post system, so there are two large, internally divided, dysfunctional parties that "represent" a lot of people whose voices are ultimately not heard and whose interests are not represented by anyone.
The parent comment was referring to mathematical properties of the voting system itself which tend to give rise to a two party system almost automatically (specifically, the combination of a first past the post voting system and single-representative districts)[1]. The Founders were not aware of these effects and could not compensate for them, so their vision of a party-free democracy did not prove relevant to how the system actually functions in practice.
Given the "rules of the game" set in US Constitution, a two-party system is a natural outcome, all other situations and fluctuations converge back to that - as shown both by game theory and practical historical evidence. Intent doesn't matter here - if you want different systemic outcomes, you have to change the rules.
But these things are not realistic with a first-past-the-post system, so there are two large, internally divided, dysfunctional parties that "represent" a lot of people whose voices are ultimately not heard and whose interests are not represented by anyone.