If this is happening, the major reason for it is the fact that some white Americans are coming to believe that their dominant position in American society is under attack.
When white Americans were completely dominant in our society, they did not need to feel that they were a racial group. It was not necessary for them to feel this because they were simply the most important group in society. They did not get treated in some generic way because of their race and therefore they did not need to see themselves in racial terms.
But now, many whites think that they are losing their dominant place in our society. All of a sudden, they have come to feel that they have interests that are based on their race. Now that they feel that they are threatened, they feel the need to come together as a race so as to defend those interests. In this way, the perceived decline in the status of whites has led to (one can argue) the rise of whiteness as a racial identity.
Many people want to have an identity to connect to. With an increasing number of various cultures coming and living in the United States one is seeing a change in the way people look.
If a person considers themselves white, it is a personal thing as to why they do so. It could be that they feel they can not fit into any other race because of how they look, act, or feel. It could be that they know their family history and want to reinforce that their family is white and want to preserve that.
People are ultimately raised in different ways and what is taught by family as well as outside influences may have an influence of whiteness being a racial identity.
This is a complex issue because there is a long history behind this.