It was Otto Warburg, a German biochemist, who in the 1930s discovered that while both normal cells and cancer cells require sugar, they have different metabolic pathways. Normal cells convert glucose into energy by aerobic respiration, but cancer cells obtain energy through glycolysis instead of using oxygen. This leads to cancer cells consuming sugar at a rate 200 times faster than normal cells. The idea then, that sugar feeds cancer cells, has been well confirmed experimentally, as detailed below at a great article from the Epoch Times. Nations that in the past had low rates of sugar consumption, such as Taiwan, typically had low cancer rates compared to America, but when a modern American level of sugar consumption occurred, the cancer rates reached US levels.
There are many ways sugar can fuel cancer, such as from obesity, by altering the metabolism, producing inflammatory responses in the body producing DNA and cell damage, and by altering the gut environment and disrupting the ecology of gut bacteria.