DNews explains why we get old and we die. It has to do with the telomeres, the edges of the chromozomes that are used as a buffer between the chromosomes and other chromozomes. Also, these telomeres shrink as the cells are diving so that, after a given time, those telomeres will vanish and the DNA from those chromosomes is not protected anymore. Changes occur, DNA degrades, aging incurs and then death.
Also, see this video about DNA and aging: