What does John Grady Cole realize about death when he sees his grandfather in the casket?

The death of John Grady Cole's grandfather doesn't just open a new chapter in his life, it also represents the passing of the Old West. When John's grandfather first built his one-room hovel on the site of the ranch back in 1866, things were a whole lot different. This was a time in American history depicted—and largely romanticized—in countless books, songs, movies, and TV shows. But the Old West, with all its old ways, has changed. Ranching is no longer a profitable business in this neck of the woods. Like just about every other kind of business, it's become increasingly concentrated in relatively few hands, leaving the little guy to get squeezed out.
This crucial stage in John's life doesn't just represent the end of an historical era but also a mythical one. The death of John's grandfather is the death of the American cowboy, that great legendary figure which has captured the imagination of millions of Americans since the late 19th century.

The funeral scene referenced in your question comes at the beginning of the novel and acts as the catalyst for John Grady Cole's departure from the ranch where he has lived all his life. The ranch is losing money, and now that his grandfather has died, John Grady's mother plans to sell the ranch. John Grady offers to take it over, but his mother refuses. She wants to move on and experience something different from ranch life. Looking at his grandfather's coffin, John Grady realizes that his grandfather's death symbolically marks the end of an era or an outdated way of life. The promise of the frontier and the dream of the wild west is over. The ranch has not been profitable for years and, as his mother has decided, it's time to move on. The rest of the story will recount his aimless adventures winding down towards Mexico.