A democracy by true definition is the right for the people to govern themselves. This is split into two versions, one called direct democracy, and the other called indirect democracy.
In Direct Democracy, each person's opinions are counted, and everybody votes directly for the laws established by the government. The majority would then rule. Examples of this can be found in Athens of Ancient Greece, in between times of upheaval.
In Indirect Democracy, each person votes for a representative of their ideals, and if elected, the official would then try to push for legislations and laws that reflect those believed by either his/herself as well as the voters who voted for them.
Now to the question. The First Amendment states that: "Congress shall make no law  prohibiting free exercise... or abridging the freedom of speech... or the right of the people... to assemble... and [allowing the people] to petition the Government [to address problems]."
As you can tell from the quote, the 1st Amendment allows the individuals to have a certain amount of rights. Starting from the smallest, is the right to exercise free speech. Each person is allowed to say what they want, though there will be consequences to what you say. The second part is to allow people of like mind and beliefs to assemble in an orderly fashion, without the government interfering. The last part is allowing the citizens to directly petition the government for changes.
Democracy is all about a person's beliefs and opinions to be heard, and for their voice to be reflected in the government. Without the First Amendment, suppression would take place. The beliefs of some would be put down and may come under flak and even oppression is voiced. This can be seen in many Middle East countries (barring Israel and Egypt to a certain extent). The protection of the First Amendment is necessary to protect a person's right to say what they want, and to take action in doing what they want to do. Without this ability, all of their rights will be in jeopardy and there will be no democracy.
