As I see it, you seem to have two questions:
1. Why do people age?
2. How do people die of "natural causes?"
I'll do my best (with my limited knowledge) to address them:
1. The general consensus is that cells replicate less quickly with each successive mitotic division. This doesn't apply to germ cells (or to stem cells, for the most part). One theory is that cells have "telomeres" that are extensions at the end of their chromosomal DNA that get successively chopped off bit by bit with each cell division. Eventually you start chopping off important DNA, the cell accumulates deleterious mutations, and it dies. Aging researchers are investigating ways of lengthening these telomeres (ie. via telomerase).
2. As we age there are a number of degenerative processes, whether it is neurological (ie. MS, Parkinsons, Alzheimers), cardiovascular (MI, heart failure), neurovascular (stroke, hemorrhage), or cancerous. (Scary fact: When you get old, almost EVERYONE has cancer. It's just that most of the time it's not in a place that messes with your biological functions). All these conditions seem to occur in people who live healthy lifestyles, and Parkinsons and Alzheimers have staggeringly high rates of incidence in the elderly. Each one of these degenerative diseases has a different etiology.
Now, you might ask, why the hell does this have to occur? The answer to this is simple: evolution. There is no evolutionary incentive for humans (or any other organism for that matter) to live indefinitely. Our biological clock has been optimized to produce children and raise them until they are old enough to have children of their own. Society has pushed back this expectation of child rearing about a decade (or more). Since evolution works by "who produces the most viable offspring," pretty much anything you do after reproductive age has no effect. Therefore, there was no "reason" for us to develop innate protections against the aforementioned degenerative diseases.
Hope this helps.