Snoring is produced by soft tissues vibrating when they flop where they shouldn't (i.e., blocking airway).
It is thus indicative of reduced airflow, which can result in negative health outcomes - including yes, brain damage. The full condition is called obstructive sleep apnoea (OSA), but there are other conditions like upper airway resistive syndrome.
From the pictures I see online, it looks like an over the counter mandibular advancement splint (MAS). MAS are used for management of OSA, but only per the prescription of a sleep physician, who needs the results from a sleep study as well as an examination of the patient.
Not all OSA sufferers will tolerate let alone benefit from a MAS. There are drawbacks and side effects with a MAS that may not be discussed with all patients, let alone purchasers of an OTC product.
I have a custom fitted one, and even as someone in the industry, I find there are certainly limitations to it.