What your text is claiming is that there was little or no sense that the thirteen colonies were all one country at the beginning of the American Revolution. At that point, the people all saw themselves not as American citizens, but as the citizens of their respective colonies. It was only the experience of the war and all its hardships (endured for a common cause) that made them into American citizens.
During the war, people from the different colonies fought side by side. As they did so, they came to see that they had a great deal in common with one another. They came to identify with one another and see themselves as citizens of a united country. They also came to have more of the attitudes that we identify with being American. For example, they demanded to be treated with respect by their officers even though the officers were above them socially and in military rank. This egalitarian attitude continued after the Revolution.
What your text is saying, then, is that Americans came to feel more united and to have more of what we would think of as American attitudes as the war went along.