Security is an illusion.
Take this to mean even security in the belief of a set of rules as protecting you or anyone else. If rules by themselves are enough then making murder illegal should solve all murder.
Protests mark civil instability. Any instability can be a potential weakness to be exploited. The instability of a protest can have far reaching effects.
They matter because they are frequently the last line of peaceful action, if they themselves are even peaceful.
A protest left unchecked quickly becomes a riot. And riots are not good for politicians. Many people judge politicians by how well they handled protests as well.
And there are countless examples of protests involving the loss of life.
Rules only matter when enough people follow them. If a protest has enough support, it can lead to a coup. Even without the protest itself being a physical threat, they are a clear political message.
A common trait among protests is that they are usually very one-sided. Support for the protest is high while opposition for a protest is usually low. A small group of people feel passionately about something that the rest of the people don't feel strongly for or against.
This means that it is typically inexpensive politically to listen to a protest. Concessions can be made with protesters often without upsetting your own supporters too much.
Countless protests throughout history have shown to have mattered. They are proven to be meaningful. Protests have also established themselves to be dangerous to the situation that produced the protest. In this sense, protests can be viewed as a symptom of an underlying problem that doesn't go away when the crowds disperse. Politicians know this.