I think this question can also be said to be - how can the Atkins diet work?
Atkins and other low carbohydrate diets sound counterintuitive. They vary in details, but the basic theme is: cut back on carbohydrates - e.g. bread, sugar, fruit, pasta, milk etc etc. But eat as much fat and protein as you like - e.g. steak, ham, bacon, eggs, cheese etc etc.
These diets really do work. Our metabolism allows us to draw food from two sources. Glucose or fat. (there is also something called glycogen, but that is only used short term and not relevant for diets long term).
It was known in the late 50s that our metabolism will to switch over consuming fat, allowing weight loss, only based on the amount of carbohydrates we consume. low carb diets initiate this change from glucose to fat metabolism because our bodies make the switch based only on the amount of glucose available in the blood. Our bodies think we are starving even if we are gorging ourselves on food which is has little or no glycemic index.
Low carb diets are not a hoax or snake oil - they really work. They can be a shock to the system and so one should discuss with your doctor before engaging them.
The first link you have is also true though. Eating fat will usually just make you gain weight. The low carb response has to be induced by basically removing all carbohydrates from the diet. If you break the low carb regime and eat a large amount of carbs, the body will switch practically instantly out of starvation mode and also take in all the fats you're eating. Lots of people on low carb diets can gain a lot of weight rapidly if they break the diet.