Fat free products often have not replaced the fat with something that is also harmful to people. Sugar free products do when they use aspartame, but fat free products usually just don’t taste as good as fat filled products. Not that aspartame tastes very good either.
A food that is advertised as trans or saturated fat free is just using a different type of fat. A food that is advertised as fat free probably replaces the fat in the cooking process with vegetable oil, or something similar that is not prejudicial to your health.
The only ingredients in fat free foods that are prejudicial to your health are the same ones that are in the foods that do have fat, and there are a great many of them.