In this case it also relies on the resistant cancer cells having altered metabolic pathways compared to your average cancer. It's not an universally applicable strategy.
Remember, fasting didn't work out so well for Jobs.
Jobs was a fruitarian in the 80s or earlier, back when he named his company "Apple". AFAIK he was neither fasting or being a fruitarian to fight his cancer, but overall he was a finicky eater all his life, a (sometimes lax) vegetarian with bouts that resembled a true eating disorder.
I don't have a source, but I recall reading that friends of his desperately tried to convince him to undergo "real" treatment for his cancer instead of relying on alternative medicine, so apparently he did try other options early on.
Didn't he only start after the diagnosis? Precancerous cells stand a better chance of being starved with fasting than cancerous cells. By the time the cell is cancerous it has built multiple metabolic pathways to power itself, making it more difficult to kill.
Why is it the worst thing you can do? It's funny - it's so easy to believe in conspiracy theories but then Shaolin monks and Steve Jobs die of cancer, and suddenly all of that goes out of the window. (Unless you make up another theory, but that's getting too convoluted)
As a guess, I'd say it's because the glucose can be used by the cancer directly as an energy source, and that the fructose can either be metabolized in the tumor directly as an adaptation, or it ties up resources elsewhere in the body that could have been better used to fight the cancer.
The lack of protein or fat in the juice means that it cannot help the rest of the body rebuild itself as the tumor is sucking up all available resources. Shipments of essential fats or amino acids are being hijacked by piratical cancer cells, and never reach their intended destinations. If there were an oversupply, some might still get through.
Each cancer is different and might respond to different treatments. I was speaking very generally. This paticular cancer reacts because of its unique properties, but the principal may work on others too.
This same diet landed Ashton Kutcher in the hospital when he tried to go method for Jobs.