Out of the loop

It does provide an advantage. One side of the headgear obfuscates vision from that eye so that they can shoot with both eyes open, but don't have to learn how to block out input from the non targeting eye. The shooting eye lens can be prescription, but the main purpose is to narrow their field of vision, providing increased focus, and essentially becoming a 3rd sighting aperture. They're legal because all of the shooters have the option to use them, and most do, because they provide an advantage. The Turkish shooter doesn't use them because he's trained a lifetime without them and has learned to ignore input from his non-sighting eye. That, combined with his T-shirt as a uniform, and his casual demeanor, is what people find appealing.