If cells in our body keep on dividing into new cells how do they ever grow old?
The only cells to grow old would be defunct cells or those who won't divide into new cells like nerve cells.
What am I missing here?
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Human chromosomes contain certain DNA regions called telomeres. These regions are shortened on each cell division and once they are gone, the cell cannot divide. Therefore an average cell in our body can divide only about 50 times.
There are indications that longer telomeres may increase lifespan, however this remains controversial. Telomeres are believed to be a method of higher organisms to prevent uncontrolled cell division and cancer, therefore artificially increasing telomere length may not lengthen but shorten lifespan because of increased cancer risks.
Also, ageing is not (only) caused by individual cells growing old. Often it is the organization between the cells that breaks down. Scarring is a good example for this. The wound is closed but the tissue inside the wound was not able to organize into functional skin tissue again but rather "patched up" the wound with comparatively unorganized tissue.
The brain is another extreme example of ageing by disorganization. It has been shown that even adult brains contain neuron progenitor cells (albeit fewer than other tissues) so even adults can regrow neurons in the brain. However, while the cell body of the neuron can be replaced, all its (sometimes thousands) of connections to other neurons are lost. Therefore, brain tissue can be regenerated in principle, but the information it contained cannot.