The president as chief citizen is supposed to be the face of the people, and represent his people as the popular leader. The president must be trustworthy and work for the public interest. The president must put the nations best interests above himself and one person or one group of people.
Here Clinton is making a speech to the people about whats currently going on.
The presidents job as Chief citizen is important because as the chief citizen he makes and builds trust with the people. He also shows the people how he expects them to act and tells them why, leading by example, is always important because those who have no idea where to go and what to do can always look at him and see where they need to be going. The presidents face is an important figure simply because the trust and feeling it brings up within people dictates how others act within the country and how those out of the country react.