I am not on the autistic spectrum, but in my experience walking does help a lot with thinking things through. Part of it is certainly the lack of distractions, but there is something about the movement too. I'm not sure why you've brought autism up.
1. The #1 example featured in the article and discussed at length — Charles Darwin — is known to have exhibited autistic symptoms, and was retro diagnosed with Asperger's by medical professionals. However, this has not been discussed well in the article.
2. Walking is a form of stimming
3. Stimming while thinking is a symptom of ASD
4. It benefits the society at large to be aware of this symptom, as it is frequently looked down upon and considered ill behavior
5. In particular regarding walking: while many people can benefit from walking while thinking, it might be required for people with ASD, like Charles Darwin. However, children in our classrooms (and most office employees) don't have that option; even fidgeting while listening is not allowed in many schools.