This type of construction is not common in normal colloquial speech. It’s very formal-sounding. It has the same meaning as
Within us, the people of the United States, a serious and purposeful rekindling of confidence is evident.
Within us, the people of the United States, it is evident that there is a serious and purposeful rekindling of confidence.
there in the original sentence is being used as a pronoun. This is similar to saying there exists- that phrase is used in predicate logic and sometimes other contexts but it always sounds very formal.