How can I lose weight if I eat a lot of fat?

The LCHF method advocates reducing carbohydrates and adding fat to my menu. How can I lose weight if I eat a lot of fat?
Clinical studies have shown time and again that fat is not fattening. Fat, including saturated fats, is very important for our bodies as it helps, among other things, to build stable cell membranes in our organs. Also, about 60% of the brain is made of fat.
The body likes to use fat as energy source, but it is easier to burn carbohydrates. So, if you eat the carbohydrates together with fat, the body will start with burning the sugars and save the fat for future use (which will probably never occur, if we continue to feed the body with carb energy).

With LCHF the amount of carbohydrates in the food is reduced so the body can burn fat instead.