The paper starts with reference to hunter-gatherer diets, and their relatively protein-rich, low-carb nature, and remarks that cancer has been found to be rare in societies eating such a diet. It then goes on to postulate several major mechanisms that may account for this association. These include:
1. Cancer cells feed preferentially on sugar (glucose) Glucose (from sugary and starchy foods) provides the prime fuel for cancer cells, so a diet lower in carbohydrate may therefore reduce tumour development or progression.
Ketones are your friend.
2. Insulin and IGF-1 can stimulate tumour cell growth High carbohydrate diets increase levels of insulin and what is known as insulin-like growth factor-1 (IGF-1) which stimulate tumour cell growth. A lower carbohydrate diet may reduce tumour proliferation as a result.
3. Ketones suppress cancer Very low carbohydrate diets can lead to the production of ‘ketones’ (mainly produced from fat) that suppress tumours.
4. Low-carbohydrate and ‘ketogenic’ diets ‘starve’ cancer Low-carbohydrate diets mimic caloric restriction and ketogenic diets mimic starvation – and caloric restriction/starvation is linked to reduce tumour development and progression.
5. Low carbohydrate diets can reduce inflammation Inflammation is believed to be a risk factor in the development of cancer, and high-carb diets encourage inflammation. Low-carbohydrate diets have been found to be more effective than low-fat ones in terms of reducing markers of inflammation.
The paper also makes the case that such diets may help better meet the nutritional needs of those with cancer.” – Dr John Briffa
You can find the “provisional” version of the article in PDF form here. In the meantime, this is (in my opinion) one of the most poignant paragraphs from the document:
In this context, it is important to note that a low CHO diet offers further possibilities to target inflammation through omission or inclusion of certain foods. Usually, CHO restriction is not only limited to avoiding sugar and other high-GI foods, but also to a reduced intake of grains. Grains can induce inflammation in susceptible individuals due to their content of omega-6 fatty acids, lectins and gluten [159, 160]. In particular gluten might play a key role in the pathogenesis of auto-immune and inflammatory disorders and some malignant diseases. In the small intestine, gluten triggers the release of zonulin, a protein that regulates the tight junctions between epithelial cells and therefore intestinal, but also blood-brain barrier function. Recent evidence suggests that overstimulation of zonulin in susceptible individuals could dysregulate intercellular communication promoting tumorigenesis at specific organ sites .
Paleolithic-type diets, that by definition exclude grain products, have been shown to improve glycemic control and cardiovascular risk factors more effectively than typically recommended low-fat diets rich in whole grains . These diets are not necessarily very low CHO diets, but focus on replacing high-GI modern foods with fruits and vegetables, in this way reducing the total GL. This brings us back to our initial perception of cancer as a disease of civilization that has been rare among hunter-gatherer societies until they adopted the Western lifestyle. Although there are certainly many factors contributing to this phenomenon, the evidence presented in this review suggests that reduction of the high CHO intake that accounts for typically >50 E% in the Western diet may play its own important role in cancer prevention and outcome.” 
It’s nice to see further evidence of what the Paleo Community has known to be true for a long time, that a lower-carbohydrate version of the Paleo Diet is one of the best, easiest and yummy ways to secure your health for the future. If you want to live a long healthy life, and stand the best possible chance of avoiding cancer, heart disease, and pretty much any disease of civilization you can think of, the Paleo Lifestyle is the way to do it.