I don't know if the claim is true or not, but there is a big difference between replacing atoms and replacing cells. Functional proteins (i.e., enzymes) are replaced at a pretty rapid rate; mostly this consists of taking the subunits apart and putting them back together, but new material must get incorporated at times, as well. The tiny amount of us that is DNA should contain some pretty old elements, as it is replicated in a semi-conservative fashion -- though even old strands are disrupted and replaced through repair mechanisms. The nonliving matrices of bones and teeth may or may not have been included in the estimates. If not, though, that leaves living cells, which are highly dynamic. Though you are stuck with a neuron for life, the material that makes up the neuron -- membrane, proteins, et al. -- is constantly recycled.
Lets take an uninjured brain. Whatever it is in there is completely replaced within a few years. Yet you are still the same person.
The structure, however, remains the same. If you change the hoses on your car, it's still the same car, doing the same things; functionally, the car has not changed.