The theory goes like this: Metabolism in all mammals is not 100% efficient. Oxygen is actually a very toxic chemical that produces free radicals, which can be very destructive. Same goes for sugar. When we break down sugar and use oxygen to generate energy (ATP), our cells inevitably generate free radicals, which is a waste/unwanted substance. The main defense mechanism against free radicals are antioxidants. If the cell lacks antioxidants (usually it does), these free radicals attack lipids, proteins, and more importantly, DNA. The cell tries to repair DNA, but the process is not perfect. Over the years, due to our own metabolism, we acquire cumulative DNA damage, and the longer we live, the more chances that some key proteins are mutated, and we get cancer. Apart from antioxidants, cell cycle regulation, DNA repair, another mechanism exists to protect us from inevitable cancer: aging.

As we get older, and accumulate DNA damage, our stem cells sense this happening, and put limits on their own reproductive potential. They slow their own cell replication to move themselves as far back from cancer as possible (fast replication). As a result, our tissue does not turnover as fast, hence our skin gets wrinkled, our immune system weakens, etc... Whether we remain young and have a high risk of cancer, or age very fast and be protected from cancer is a balance intrinsic to each and every one of us. Dogs get cancers at a very early age because their metabolic rate is incredibly fast, and they develop DNA damage much faster. However, your lifestyle can edit that balance. If you have a healthy diet low in sugar, high in antioxidants, in theory you can reduce DNA damage, keeping your DNA looking younger, and reducing risk of cancer when you are older, and prevent aging at a young age. Other things you can do is a long topic of discussion, but generally eat less sugars, eat "healthy", have a low basal metabolic rate, avoid radiation exposure, have good DNA repair genes, and don't stress (cell stress response).