(Note: I have chosen in this answer to avoid the controversy surrounding the "Founding Fathers of the USA", who were certainly not of one mind, as choster has already explained in his fine answer, and stick to the US Constitution itself, as it was ratified and amended.)
Excellent question IMO, and there does indeed appear to have been something of a shift at some point:
Indeed, there is no mention of the term "Democracy" in the US Constitution or Declaration of Independence, or even an explicit provision for what we call "the popular vote" for President or for Senators. In the original body of the US Constitution, the election of the US President is given to the Electoral College, an institution that persists down to the present day, and the seating of Senators is delegated to the State legislatures:
Article II, Section 1:
The executive Power shall be vested in a President of the United
States of America. He shall hold his Office during the Term of four
Years, and, together with the Vice-President chosen for the same Term,
be elected, as follows: Each State shall appoint, in such Manner as
the Legislature thereof may direct, a Number of Electors, equal to the
whole Number of Senators and Representatives to which the State may be
entitled in the Congress: but no Senator or Representative, or Person
holding an Office of Trust or Profit under the United States, shall be
appointed an Elector.
(Somewhat modified by Amendment XII, but the Electoral College remains)
Article I, Section 3:
The Senate of the United States shall be composed of two Senators from
each State, chosen by the Legislature thereof, for six Years; and each
Senator shall have one Vote.
These provisions reflect the view expressed by Hamilton in Federalist 68:
It was equally desirable, that the immediate election should be made
by men most capable of analyzing the qualities adapted to the station,
and acting under circumstances favorable to deliberation, and to a
judicious combination of all the reasons and inducements which were
proper to govern their choice. A small number of persons, selected by
their fellow-citizens from the general mass, will be most likely to
possess the information and discernment requisite to such complicated
investigations. It was also peculiarly desirable to afford as little
opportunity as possible to tumult and disorder. This evil was not
least to be dreaded in the election of a magistrate, who was to have
so important an agency in the administration of the government as the
President of the United States. But the precautions which have been so
happily concerted in the system under consideration, promise an
effectual security against this mischief.
Contrast this view with Classical Democracy as exemplified by Ancient Athens:
A political system in which the people do not elect representatives to
vote on their behalf but vote on legislation and executive bills in
their own right.
"Despised" is perhaps too strong a term, but it's clear that Hamilton and his constitution did not embrace classical democracy with open arms, preferring rather a "representative republic", closely resembling the bicameral structure of the British Constitutional Monarchy with "Upper and Lower Houses" - Lords and Commons - and the ancient Roman Republic in some respects.
It is difficult to pinpoint an exact moment or event that triggered the shift towards "Democracy", but a good starting point would be the 17th Amendment to the US Constitution, ratified in 1913, although not agreed upon by all the States.
The Senate of the United States shall be composed of two Senators from
each State, elected by the people thereof, for six years; and each
Senator shall have one vote. The electors in each State shall have the
qualifications requisite for electors of the most numerous branch of
the State legislatures.
States that rejected Amendment XVII
The following states did not ratify the Seventeenth Amendment: Utah (explicitly rejected)Florida Georgia Kentucky Mississippi Rhode Island South Carolina Virginia.
Electing Senators directly rather than through State Legislatures is an obvious and dramatic shift in emphasis from the power of the States acting as proxies, to the direct representation of individuals in government - something akin to "Democracy".
However, the debate regarding the method for seating Senators had been raging for quite some time previous:
Roots of Amendment XVII
However, over time various issues with these provisions, such as the
risk of corruption and the potential for electoral deadlocks or a lack
of representation should a seat become vacant, led to a campaign for
reform. Reformers tabled constitutional amendments in 1828, 1829, and
1855, with the issues finally reaching a head during the 1890s and
1900s. Progressives, such as William Jennings Bryan, called for reform
to the way senators were chosen...
If we choose, we might go back further in time to Amendment XIV, ratified in 1868:
Section 1. All persons born or naturalized in the United States, and
subject to the jurisdiction thereof, are citizens of the United States
and of the State wherein they reside. No State shall make or enforce
any law which shall abridge the privileges or immunities of citizens
of the United States; nor shall any State deprive any person of life,
liberty, or property, without due process of law; nor deny to any
person within its jurisdiction the equal protection of the laws.
One the most important goals/impact of this amendment was to ensure enfranchisement for all former slaves in the post Civil War era. One could argue that this amendment caused the American political consciousness to focus more on the power of individuals in government and elections as opposed to government 'by proxy'.
We might also cite Lincoln's Emancipation Proclamation of 1863, which freed the slaves in the rebelling states of the Confederacy, and the victory of the North in the American Civil War, as factors which consolidated Federal power and diminished "States Rights" - the sovereignty of the individual states as opposed to the Federal government. Again, this made the individual more of a focal unit of power in the American Republic, than the States: a shift towards "Democracy."
This issue of States' Rights vs Federal power again goes back to pre-Consitutional days, was central to the American Civil War (Slavery and the vehement debates surrounding it that eventually resulted in war, were the result of the power granted to individual states to determine their own laws regarding slaves.) and has been an important theme throughout American history down until the present, with the general direction always being away from emphasis on States' Rights. This is manifest most dramatically (aside from the victory of the North in Civil War) in the many US Supreme Court decisions invoking the Commerce Clause and the General Welfare Clause to trump "States' Rights", and again effectively making the individual the most important unit of power in the electoral process, rather than the States (This is indeed a very lengthy discussion which I will not get into here.)
Along the same lines as Amendment XIV, we can cite Amendment XIX, ratified in 1920, which gave women the right to vote. Again, the power of the individual in the electoral process becomes a focal point.
The right of citizens of the United States to vote shall not be denied
or abridged by the United States or by any State on account of sex.
There have also been numerous movements and attempts to eliminate or circumvent the Electoral College and move to direct Election of the President. This debate continues down to the present and was highlighted in the contested Election of 2000, when election results in Florida were disputed and that election's outcome came down to the question of which candidate could claim the electoral votes of Florida, although there was no question that Al Gore had won the majority of the national "popular vote". See Ideological Endowment: The Staying Power of the Electoral College and the Weaknesses of the National Popular Vote Interstate Compact - one example from a law journal culled from hundreds of references on this issue.
Again, it is difficult to pinpoint the exact time, events and reasons for these apparent shifts, although none of the issues are really new - they date back to Constitutional times, as we see from Federalist 68, the sources that choster has cited in his erudite answer, and many other sources.
My personal impression regarding "the Big Picture", taking into account all the above mentioned constitutional/legal shifts, is that as (virtually) universal literacy, education and suffrage became the norm in the USA, communication improved and more and more emphasis was placed on individual rights and liberties in the post Civil War era etc., as explained, Americans at large felt that they were all entitled to a direct say in the electoral process.
The result of this gradual change in American political focus and consciousness? Something closer to Athenian style "democracy" has become the projected image of the American Republic, and the term "Democracy" in modern American English is virtually synonymous with the representative republican form of government of the United States.
Still, that form of that government, in spite of the changes we have mentioned, remains quite different from that of Classical Democracy, as exemplified by Ancient Athens.