This is a very good question with a very complex answer, because scientists are still trying to figure out why we age!

The simple answer is that as we get older, the cells in our bodies stop working as well. We may end up with fewer immune cells, for example, which affects our ability to fight off pathogens like cold and flu viruses - this is why the flu is much more dangerous in elderly people.

However, that doesn't explain why our cells stop working well in the first place.

To understand why we age, we have to look at aging from an evolutionary perspective. After all, shouldn't there be an evolutionary advantage to living for hundreds of years? It seems obvious that living longer would be an evolutionary advantageous trait - if Fred lives to 30 and has 1 child every 5 years after he turns 20, and Bob lives to 100 having offspring at the same rate, wouldn't Bob end up with more children to spread his long- living genes?

Fred = 2 kids, Bob = 16 kids

However, in biology, all traits have their tradeoffs. You always have to examine these traits in the context of their environment. For example, anteaters have long claws that allow them to dig into anthills in order to get more ants. This is a beneficial trait for an anteater, but can you imagine humans with long claws? We'd have little use for them, and they would most likely get in the way of anything useful we try to do! It's possible that genes causing old age have downsides as well, and that the benefit:cost ratio isn't strong enough to promote their survival.

Additionally, we have to understand that evolution only works under selection pressures. In the wild, it is rare for organisms to live until old age. Most animals die due to starvation, being eaten by other animals, injury, or disease. Humans have only recently (in terms of evolution) figured out how to live consistently until old age - therefore, there was likely not much selection acting on older people in our evolutionary past.

Finally, one biological explanation for why we age is that animal chromosomes shorten each time a cell replicates. We have evolved short "caps" on the end of our chromosomes called telomeres, which are regions of extra DNA that protect genes from being degraded. However, with enough cell replications, these telomeres are degraded too much and important genes are affected. Cells lacking telomeres on one or more chromosomes usually stop replication, a process referred to as senescence.

You can see a nice picture on the pdf version of this answer:
link