How do they do what?
Shaking their head? Just like that: up and down.
The head shaking syndrome most often caused by allergies, dust, or bugs. The net is attached to the noseband and covers the nostrils (it's long enough and shaped to cover nostrils/upper lip) providing some kind of barrier. If you read reviews on web number of people say it helped (and I say so too from my personal experience).
BTW, it has nothing to do with training, but rather with health.