Good question. A lot of people want to know the answer.
Yes, our cells divide and produce more cells to replace the ones that die. But this is all directed by the information stored in our DNA.
You can imagine that as the DNA gets copied over and over from the original template present in the egg+sperm, some mistakes might occur and the integrity of the genetic material will generally degrade. This is generally the basis of why our bodies start to fail on a cellular level (e.g. cancer). Even though we produce "new" cells, the DNA has already been copied over and over and subjected to external sources of damage such as UV radiation or harmful chemicals in the environment... think of making a copy of an original document on a traditional scanner, and repeating that over and over.
In mammals and other eukaryotes, DNA is arranged into structures called chromosomes, capped by a structural component called telomeres. These are kind of like tape on the end of a rope that keep it from coming undone and maintain it's structural integrity. The more times a cell has replicated, the shorter these telomeres become, because the cell can't replicate the DNA to the very end of the strand. Once the telomeres reach a certain "shortness" it signals the cell to die... effectively limiting the number of times a cell can divide. It is generally believed that this also has some effect on lifespan.
Also, some cells do not replicate, such as most neurons. So once you maintain damage to your brain it can't be easily repaired, especially considering the complexity of neural networks that you have formed over a lifetime. And since your brain controls many other bodily functions, once you lose that you're out of luck.
You can think about this on the level of the cell or the level of the organ or the level of the body as a functioning whole... but either way it all comes down to the same theme of a eventual reduction in the integrity of the system that can no longer be repaired.