In communities with a large population, representative democracy allows the people to be heard without requiring everyone to submit their opinion on a subject individually. This means that more people get to have a voice without the crippling slow-down that would result from having thousands or even millions of individuals vote on each individual issue.
Democracy has been around for thousands of years, and through that time many different forms of democracy have been in place. In small towns some of the older ways are still in play through the use of town meetings and per-issue balloting. While these do truly give every citizen a voice directly in the process, as the population grows, this becomes a burdensome task that prevents progress from being made.
In theory, representative democracy allows a group of like-minded people to send one of their own people to vote in their interest, thus representing them while being able to progress at an acceptable pace through the business of the government.
There is some rightful criticism on this issue when it comes to extremely large bodies of people, such as seen in Congress, where one representative is required to represent millions of people by himself. This has caused some groups to call for almost all federal powers to be handed over to states, and even a select few suggesting that the nation do away with the federal government all together, citing that they cannot possibly represent all citizens.
Yet, this system remains the best option on the table, providing the most representation to the most people. It remains an issue considered quite frequently by politically minded people throughout the world as they strive to come up with the best form of democracy.