Why does Hydroponics work so well?

That's simple. If you give a plant exactly what it needs, when it needs it, in the amount that it needs, the plant will be as healthy as is genetically possible. With hydroponics this is an easy task; in soil it is far more difficult.
With hydroponics the plants are grown in an inert growing medium (see below) and a perfectly balanced, pH adjusted nutrient solution is delivered to the roots in a highly soluble form. This allows the plant to uptake its food with very little effort as opposed to soil where the roots must search out the nutrients and extract them. This is true even when using rich, organic soil and top of the line nutrients. The energy expended by the roots in this process is energy better spent on vegetative growth and fruit and flower production.
If you grow two genetically identical plants using soil for one and hydroponics for the other, you will almost immediately see the difference this factor makes. Faster, better growth and much greater yields are just some of the many reasons that hydroponics is being adapted around the world for commercial food production as well as a growing number of home, hobby gardeners.