Antiquity (that is, the societies of ancient times) and England itself were two of the principal inspirations for American ideas about liberty, independence, and the form that the Founders envisioned for their new government.
The concept that historians now call republicanism derived from a number of inspirations, notably the societies of ancient Greece and Rome, where all citizens (with the notable exception of slaves, who were not considered citizens) voluntarily subordinated their private interests to the common good. These participatory republics predicated their stability and authority on the virtue of the citizenry as a whole and their resistance to corruption or tyranny. For eighteenth-century republicans, a person with "virtue" owned property, possessed an intrinsic sense of morality, and was willing to subordinate his own interests for the interests of the community; the public good. These were the only sorts of people whom Founding Father Benjamin Franklin thought capable of freedom. Republican government was, by design, antithetical to monarchies or aristocracies in which a rigid hierarchy predetermined the social structure and a small number of powerful people ruled over the masses with little to no oversight. Yet both Greece and Rome utilized slave labor and denied women any direct voice in government; these paradoxes (human equality amidst gender discrimination, liberty coexisting with bondage) would also carry over into the United States.
Republicanism, the ideology of the American revolutionaries, was not the same as democracy. It would be understandable if this is the part that starts confusing you; but a democracy and a republic are two different forms of government. This country started as a republic; the question of how democratic it has since become is an open-ended debate that will carry on for some time. So let's focus on the history. America was not formed as a democracy in the classic sense, and the word itself does not appear anywhere in the Constitution. From the Greek demokratia, or "rule of the people," a democracy initially indicated a government in which the citizenry directly participated in the function of government; the only time such a government ever existed was in 508 B.C. in Athens, and even then it only encompassed a democracy for males over the age of eighteen. Even at the time of the Revolution, America was too sprawling and populous a country to entail such direct participation on the part of every eligible voter, even when the suffrage was limited to white, propertied males. How could everyone possibly participate in the same town meeting to make the decisions that affected everyone? Instead, people elected representatives to presumably speak and act on their behalf. Besides, many Founding Fathers did not want a pure democracy; they thought that some sort of executive power was necessary, along with an "upper house" of the government to serve as a sort of "buffer" between the masses and the powers that governed them.
So the American system emerged as a sort of uneasy hybrid between democracy and republicanism. One result of its long-term success has been a redefinition of the terms so that our modern-day meaning of democracy shades into republicanism, and vice versa. We don't have to see them as opposing systems, but the question of whether this is or is not a democracy has propelled many social movements to reform the electoral system, ensure the freedom of the press, and make elected officials more accountable to the people and less reliant on campaign donations from lobbyists and other special interests.
The definition of democracy has expanded to encompass a system that more closely resembles America's; that is, a rule of the citizenry through its elected representatives. Yet this rule applied only in the House of Representatives; the Senate was elected by state legislatures until 1913, and the Electoral College still mediates the popular election of the president. Only a sub-section of the American citizenry elected the House and the president; not only were voters required to be male and white in order to be eligible, but several states passed property requirements under the presumption that voters could only be trusted to act in the best interests of the country if they had a vested stake in the country and the means of acting independently, via land ownership.
Despite these considerable restrictions, the United States became a wholly new system of government in a world primarily dominated by rigidly hierarchical, aristocratic systems. This new system was rooted in the English tradition of open debate and the struggle to ensure and protect the rights of Englishmen. The British faction known as the radical Whigs spoke out against the perceived threat to liberty posed by the arbitrary power of the monarch and his ministers. Colonists read their attacks on the king's ministers and their use of bribery and patronage. Many colonists believed the Whigs when they warned that such corruption must be vigilantly guarded against to prevent the decay of the entire society. When Britain tried to extract more revenue from the colonies after 1763 to pay for the expensive French and Indian War, while attempting to assert its authority by restricting western settlement to the area east of the Appalachian Mountains, the colonists were both disgruntled and alarmed by the fulfillment of the Whig prophecy: an affront to their rights. When the stirring words of radicals like Tom Paine combined with the angry denouncements of King George III, the path was cleared for a publicly supported revolution against a monarchy that was deemed corrupt and tyrannical beyond repair.