How did the development of the American attitude about who we are as a person affect the growth of the West ?
Different people could give different answers to this. Please check your text or your class notes to see if your instructor inspects you to give some particular answer. One way to answer this is to say that the development of American ideas about who we are as people affected the development of the West by making it happen more quickly and by making it happen in a way that was very bad for the people who already lived in the West.
American attitudes about who we are as people tend to center around the idea that we, as a group, are a special people and the idea that we, as individuals are self-sufficient and independent. The idea that we are and should be self-sufficient and independent helped to hasten the growth of the West. Because Americans felt this way, many of them wanted to leave the East and move to the West. They felt that they could be their own bosses in the rural West, which was more in line with ideas of what Americans should be than working for someone in a company in the East was. Because Americans came to believe that they, as a people, were special, they also felt that it was their right to take land from the non-American peoples who lived there. This attitude of superiority allowed them to think that it was right for them to forcibly take land from the Native Americans. Because of this, the growth of the West ended up being very bad for the Native Americans (and for others, like the Hispanics who had lived in what had once been northern Mexico).