When I tell people what I do, they're not real sure. In a nutshell, the act of public relations is "the professional maintenance of a favorable public image by a company or other organization".
That means, through communicating through a variety of mediums, you are simply letting people know what you're doing and why it matters to them. Through social media, newsletters, e-mails, events and many other ways, sharing with your customers and constituents makes them feel important and invested in your organization.
So, you ask, why is it important to communicate with your publics? Because if you don't tell people what you're doing, they won't know. And if you want to be a part of their lives and make them part of your success, you need to communicate with them.