The term "graying of America" refers to the idea that America has, since the 1960s, been steadily growing older. During the '60s, there was a huge boom in the number of young people as the "baby boom" of the post-WWII years became young adults. This helped to lead to the youth-obsessed culture of the '60s and to things like the Counterculture.
Since then, however, America has been getting older "graying" as the baby boomers age. This phenomenon was remarked on as early as 1973 in the Time Magazine article in the link below. As the baby boomers have aged, American society has become somewhat more focused on the issues and needs of older people.
This is what the graying of America is. It is both a change in demographics (the average age of the population rising) and a change in society (a move from a youth-centered society to one that is more focused on the needs of older people).