1. Bigger animals have larger cells, so the number of cells does not scale linearly with the mass of the animal.
2. Cells from larger animals are more resistant to cancer development in various ways (eg more copies of TP53, more propensity to die after DNA damage).
3. Large animals have non-linear metabolic scaling. So a cell in an elephant sees fewer metabolic cycles per unit time than a cell in a mouse. Some of this is due to the greater thermal efficiency of large animals.
Now that we have more detailed data on animals, I think this paradox is a bit less interesting, albeit generally true. For example, there are some animals that just don't seem to get cancer (eg the blackbuck and the Patagonian mara, from several hundred autopsies each) [1]. Cancer is also very rare in Zebras. This is fundamentally interesting because it shows that being free from cancer is possible in an otherwise normal looking mammal. In theory therefore, we could consider engineering humans to be cancer free. To me, given the high incidence of cancer, the progressively increasing incidence in an ageing population and the incredible amount of misery it causes, this is the most ethically palatable reason to consider widespread genetic modification of humans. I think all anti-ageing efforts will ultimately bump up against cancer as the limiting factor. The technology to safely genetically engineer humans is nowhere near ready for prime time.
The association between height and cancer is not thought to be due to 'more cells'. It only applies to some types of cancer and not others. It is believed to be due to growth factor signalling eg growth hormone and insulin-like growth factors.