You shouldn't eat no carbs. Eat vegetables and fruit (less fruit then veggies ideally due to sugar content) and whole grains. Avoid fruit juices and try to eat the whole fruits instead, it's more filling and there isn't any sugar added. The main problem is that the average American eats too much sugar and simple and processed carbs. Once you've filled your glycogen stores in your muscle and liver you start to store the rest of the glucose you eat as fat. When you use up your available glucose and then your glycogen stores, your body will start breaking down your muscle protein for amino acids (which can be converted to glucose) or it will access your fat stores to slowly breakdown triglycerides into molecules that can be used for energy (ketones and acetyl CoA).
In summary: It isn't no carbs, it's the right carbs. Not no fat, but the right fat. And finally, having good quality protein.
p.s. if you want to feel full, eat foods with more fibre.