I try to use native plants when possible for the following reasons:
1) Native plants are adapted to local conditions
Native plants are part of a local plant community that is able to survive local conditions (temperatures, rain, wind, soils, insects) without any help. So if we put them in the garden they are low maintenance and reduce pesticide, fertilizer, and water use.
2) Native plants give a sense of place
Native plants make the garden resonate to what is special about the local climate, geology, and ecology. Why not celebrate that and invite the wildlife into the yard.
Hopefully, I can re-create some of the patterns I see in nature when I garden by using local materials -- plants, rocks, and wood -- to provide shade, windbreaks, and beauty.
The flip side to native plants is exotics that have become invasive. When you see trees strangled in ivy or gorse choking out everything else on a hillside, you realize that bringing in new plants to the local ecology can have unintended consequences.
I think well behaved non-natives are fine. We just have to be careful and avoid the invasive ones if possible. Because once they come in, it's hard to get rid of them.