The only way to lose weight is to take in fewer calories than you use, which for most people means eating less and exercising more. Most medics recommend a balanced diet, paying particular attention to cutting down on fat and carbohydrate, which tend to be the highest-calorie items in the diet. Some people (sometimes for medical reasons, sometimes for reasons of preference) concentrate on cutting either fat or carbohydrate to the bare minimum.
If you cut down the carbohydrates too severely you need something to fill you up so the answer is protein. The best proteins to have are the low fat ones eg fish and poultry, but we are all human and need a bit of variety, so there is a greater tendency for high-carbers to eat high-fat items.
So how can you manage to lose weight while not restricting the fat? People certainly seem to do so, and there seem to be two reasons for this:
1 A number of fats are associated with carbohydrates so, for instance, if you are not eating much bread you will probably reduce your butter consumption without needing to consciously try to do so.
2 A diet which is so lacking in variety will probably result in eating smaller quantities unless you are one of the rare people who can eat huge steaks with very little else to help it down.
I think it is better for the general health to reduce both carbs and fats a little rather than concentrate on just one or the other, and I think most doctors (apart from those who have made their fortune writing diet books) would agree. Too much fat is not good for the heart, and there is also a theory that too much protein can be harmful to the kidneys. But anything a bit extreme makes a better news story than celebrities eating a healthy balanced diet and taking a moderate amount of exercise!