"The only thing we [Americans] have to fear is fear itself," Roosevelt remarks.
He's behaving like a parent, aiming to alleviate the public's anxieties about the state of the country in the same way that a parent would quiet a child's fears.
By employing the word "we," he avoids a patronizing tone that might otherwise be implied. This phrase implies that the American people should be terrified solely of letting their dread obstruct their attempts to address the country's economic difficulties.
Before they can take physical action, they must overcome this psychological hurdle.
When the president states that the obstacles ahead aren't so difficult that they won't be able to conquer them, members of the audience should feel compelled to put aside their fears about what will happen next and focus on supporting others and the country's leadership.