For a plant to grow all it needs is water, air, light and nutirents such as potassium, phosphorous and nitrogen. Usually the plant gets these from the soil, but they can get it from water with added nutrients.
Gardeners say that hydroculture is very precise because it means that you add the nutirents to the plants, so you know exactly how much of a given nutrient they have had. With soil, even if it is tested, it can be quite difficult to know how much of a nutrient a plant is getting.
They also indicate that the risk of disease is minimised since disease can often be transmitted by soil. But ultimately, many gardeners still like to feel the soil between their fingers, so it is not a widespread practice.