Why do Xerophytes have extensive and well developed root systems ?
First we must clarify that xerophytes include many different plants. However, a commonality amongst these plants is that they have adapted to live in environments where the loss of water through evaporation and transpiration outweigh precipitation (think of an unbalanced water cycle) Root systems will vary depending on which type of xerophyte is being discussed.
Let's take a look at a catus first (probably the most well-adapted xerophyte). It's root system is very shallow due to the fact that when it rains in the desert most water will not have time to seep into the ground before evaporation occurs; therefore shallow roots stand a better chance than deep roots to absorb water. But just because the roots are shallow doesn't mean they aren't extensive/well developed. They spread out laterally from the plant center quite a ways. They are not usually very thick. They are fine with many root hairs. This in combination with a large radius allows the plant to absorb more water due to increased surface area.
Another type of xerophyte is chaparral. This is a desert plant as well and is thought of traditionally as an herb. It has a rich history with native americans. The root system of this xerophyte is both wide-spread and deep. While, the deep root system allows it to maintain stability, it also allows the plant to absorb water deep below the surface. This particular plant can live over a year without any preciptation falling to the ground.
The extensive root systems are just one adaptation of xerophytes. In order to live in arid enviroments there are many other adapations that work in concert with the roots.