Recent events address this issue well, I think. The pastor in Florida who recently wished to burn the Quran on 9/11 was as controversial as they come, and about as unpopular, but he had a first amendment right to express his feelings in that way, and the Courts over time have been very staunch defenders of the minority's right to practice free expression. In my own state, the courts ordered that a white supremacist rally not only be allowed to take place, but that it be guarded by state troopers to ensure it could happen without incident.
Why this is important for our democracy is that these minorities, political, ethnic or religious, are protected by free expression, otherwise they would be continually subject to the will of the majority, and their rights could be easily taken away. Democracy, the way ours is set up, guarantees legal rights and equality to all citizens, regardless of background or creed. While that doesn't always happen, 1st Amendment free expression helps us come close.
In order to have democracy, people need to be able to express themselves freely. A country cannot be democratic if the people do not govern themselves. If you are not allowed to say what you think, there is no way that you can truly participate in governing yourself.
Imagine a parent who tells their kids "you can help make the rules for yourself. But if you argue for some rule that I don't like, I'll punish you." Is this fair? Would the kid truly be able to help govern themselves? No way.
In a democracy, people need to be able to freely say what they think is best. They have to be allowed to express any point of view, not just ones that the government approves of.