Early office lighting was gas lamps, later carbon filament incandescent. These both gave off way more infrared and ultraviolet light than modern fluros and LEDs. A gas lamp is basically a fire. You know how you can feel the heat coming off a campfire from meters away? That’s infrared light, you can’t see it, but it still damages you. Gas lamps gave off not as much as sunlight, but sitting for 8-10 hours a day In a room full of them was enough to sunburn your eyeballs. The green visors worked sort of like sunglasses, blocking UV and blocking direct line of sight to the lamps. It was more a kind of PPE than anything else. Then it became a distinctive trait of bookkeepers and accountants.

Why just a headband and more a full hat? Maybe for more airflow, maybe because it was unprofitable to wear a hat indoors.