The question is poorly stated. The Founding Fathers were not all of one mind on many subjects— the Federalists saw danger in direct democracy, whereas the Anti-Federalists did not. Additionally, popular usage of terms like "democracy" or "republic" is quite different from a political scientist's use of such terms— indeed, quite a lot of things "don't mean what you think they mean," from liberalism to imperialism. Moreover, very few shifts in history can be traced to a particular "moment"; the world-historical figure as a character for study has been distinctly out of fashion for some decades. So, I will limit the discussion to the Federalist position on direct democracy versus representative democracy, i.e. a republic.
The entirety of The Federalist #10, by James Madison, is devoted to the question of direct democracy (which he calls "pure democracy") as opposed to representative democracy (which he calls a "republic"). It is arguably the most famous of The Federalist Papers. Madison argues that society is undone by faction, which he defines as
a number of citizens, whether amounting to a majority or a minority of the whole, who are united and actuated by some common impulse of passion, or of interest, adversed to the rights of other citizens, or to the permanent and aggregate interests of the community.…
A faction can be a minority, but also a majority, and the majority can work against the good of the whole by abusing its power. The majority simply cannot be trusted not to oppress the minority, given the opportunity.
If the impulse and the opportunity be suffered to coincide, we well know that neither moral nor religious motives can be relied on as an adequate control.…
From this view of the subject it may be concluded that a pure democracy, by which I mean a society consisting of a small number of citizens, who assemble and administer the government in person, can admit of no cure for the mischiefs of faction. A common passion or interest will, in almost every case, be felt by a majority of the whole; a communication and concert result from the form of government itself; and there is nothing to check the inducements to sacrifice the weaker party or an obnoxious individual. Hence it is that such democracies have ever been spectacles of turbulence and contention; have ever been found incompatible with personal security or the rights of property; and have in general been as short in their lives as they have been violent in their deaths. Theoretic politicians, who have patronized this species of government, have erroneously supposed that by reducing mankind to a perfect equality in their political rights, they would, at the same time, be perfectly equalized and assimilated in their possessions, their opinions, and their passions.
As an alternative, he offers
A republic, by which I mean a government in which the scheme of representation takes place, opens a different prospect, and promises the cure for which we are seeking.
The two great points of difference between a democracy and a republic are: first, the delegation of the government, in the latter, to a small number of citizens elected by the rest; secondly, the greater number of citizens, and greater sphere of country, over which the latter may be extended.…
The first allows that a body of citizens expressly chosen to refine the laws would be more focused and responsible in pursuing the true common interest over temporary or personal considerations. The second allows that direct democracy becomes unwieldy for large countries, but a republic would be efficient enough to govern while still encompassing a large number of interests, reducing the power of any individual faction.
Just read Federalist #10. The whole thing.