There are many reasons to do this.
First, straw is a very good insulator and utilizing it will make a very efficient, high-performance home.
Second, straw is a plant-based material (a waste product of grain farming) that is widely available, making it a great choice for those wishing to lessen the ecological impact of their home.
Third, straw bale homes are a pleasure to live in. The combination of straw bale and clay or lime plaster (the material that covers the bale), is not only healthy for the environment, but it makes for a very healthy and pleasurable interior environment for those living in a straw bale home. The interiors are free of harmful chemicals, they manage humidity very well, and they offer a softness not found in most other building materials.
Lastly, straw bale construction lends itself to having the owners participate in the building of their home. Workshops, like those offered by Dirt Craft Natural Building, give people the tools and skills necessary to take an active role in the building of their own home.