I occasionally do competitive matches with handguns (USPSA in the US), so I have some background knowledge. The eye cover is to reduce strain / fatigue from squinting, however it also helps the other eye focus on their aperture sight naturally, thus making it easier to take a more accurate shot (with all of the science behind closing apertures and front sight posts, linked in this thread).
In USPSA, while we don't use eye covers in a match, but a lot of competitors will use them on their dominant eye to practice fast target acquisition, independent of their pistol red dot or iron sights. Translates to very refined muscle memory when you don't need to rely on your optics to align your aim.
In most type of shooting, you're generally NOT closing either of your eyes while aiming. It's counter productive to your situational awareness
In USPSA, while we don't use eye covers in a match, but a lot of competitors will use them on their dominant eye to practice fast target acquisition, independent of their pistol red dot or iron sights. Translates to very refined muscle memory when you don't need to rely on your optics to align your aim.
In most type of shooting, you're generally NOT closing either of your eyes while aiming. It's counter productive to your situational awareness