Seabiscuit is famous because he played into an American ideal at a time when Americans needed feel-good stories. In addition, we have to remember that he lived at a time when horse racing was a much more popular sport than it is today.
Seabiscuit was an underdog. He had good bloodlines, but did not distinguish himself in his early career. Then, seemingly out of nowhere, he started to win race after race. Because of this, he came to have the image of a "rags to riches" underdog. This was in the middle of the Great Depression when America was in terrible economic shape and people needed happy stories to help improve their morale. Seabiscuit was a perfect fit for the time--an underdog horse who fought back from adversity and made good.