Turner saw the Western Frontier as being a defining characteristic of Americans. It kept Americans exploring and pushing farther away from Europe in the East. The farther West Americans pushed (the farther into the Frontier they went) the more unique of an American identity developed.
Turner saw the closing of the Frontier as an end to an era of expansionism and development of American’s character. Since he saw the ability to move further into the frontier as a developmental factor (much like a process of evolution) to American Culture and identity he felt that the closing of the Frontier threatened the development of an American identity.
Here is an article that talks a little more about the article and analyses weather Turner’s fear was justified: http://www.digitalhistory