How is the United States a representative democracy?

To answer the question, let's first define the important parts:

A Democracy in its most simplest form means a system of government where individual citizens have a direct [and meaningful] vote in their government. There are many different kinds of democracies - variations on what is a citizen, who can vote, and who or what they vote for. However, the overarching characteristics of the democracy is that at least a majority of citizen have the power to vote, that they have a choice when voting, that their votes dictate the direction of government policy, and that the citizen is (reasonably) frequently called upon to vote on government matter.

A Representative form of government is one where (some) citizens chose another citizen to make political decisions for them. That is, rather than a citizen being directly involved in politics and government, they chose a proxy to make their choice for them. Representative forms of government often have multiple layers of this action - a proxy is chosen at (say) the village level, then the village proxies get together and chose a county representative, who then may chose a regional proxy. How these representatives are initially picked depends on the actual system of government being used - voting is common, but not by any means universal.

Combining these two definitions, you find that a representative democracy is a system where the majority of citizens vote for a proxy to represent them in government - such representatives then govern in the citizen's name, and are (at least occasionally) forced to go back to the citizen to be re-authorized (i.e. re-elected) to hold their position.

The United States is a multi-faceted Representative Democracy: we elect representatives in many aspects of our government form, all of which have specific duties and limits on power. We periodically have elections to either retain or replace our representatives. And a majority of the population is allowed to vote (though sadly, less than half does).