In general, it's not really productive to try to cut fat intake to such incredibly low levels. Although you need to have at least a minimal level of protein, carbohydrates, and fat in your diet for general health, a larger amount of fat (same as a larger amount of carbohydrates or a larger amount of protein) will not necessarily result in fat deposited on the body. That is more a factor of how many calories you burn versus how many you consume.
Making changes to your diet such that you replace more of the saturated fats with unsaturated fats, baking instead of frying meat, etc. can be beneficial. But even with 15-20 grams of fat a day, if you're eating more total calories than you're burning, you'll end up gaining weight. At a certain point, trying to drop fat intake lower becomes counterproductive, as many non-fat or low-fat versions of things are...loaded with sugar instead to make them taste better. Sometimes the non-fat version of something actually has more calories in it than the normal version. This becomes even more of an issue if you eat 3 of the non-fat cookies to satisfy a craving you could fill with 1 of the cookies made with real butter.