There are many reasons for eating grass fed beef. The most common is good taste, but there are also environmental and ethical considerations.
If you are concerned about your familys' health and the quality of the food you eat, find a local farmer who raises his own food natually and is in harmony with nature and buy from him.
Rumminants (cattle and sheep the primary domesticated species) have the ability to turn forages (grasses and other forbs) into nutritious and good tasting milk and meat. Simple stomached animals like pigs and chickens are fairly efficient at converting grain into meat while ruminants are quite inefficient in converting grain. (Pigs and chickens require two pounds or more of grain to produce a pound of meat - cattle require four to five pounds or more of grain per pound of gain). Even simple stomached animals are competing with humans for a limited amount of grain available worldwide. Meat is a nutrient dense food, but it is often more efficient in feeding larger numbers of people by having them eat the grain directly.
A good portion of the earths surface does not lend itself (either because of weather or terrain) to growing grains. These same areas (particularily if managed properly) can and do produce large quantities of nutritious forages. As you can see, cattle have the ability to produce good tasting and nutrient dense food for humans that would not be useful to them otherwise.
Some people have compared grass fed beef to the production of wine. Wine experts claim that the same grape plants growing in different soils and climates will produce different tasting wines. This allows for a wide range of flavors and distinctive tastes. The same is true with forages and other plants - the animals that eat them will produce meat that is as varied and diverse as their diet.