The classic snooker player visors you are probably thinking of are from early to mid-1900's and are called "green eyeshades" or "green eye-shades" Apparently, green eye shades were usually made from transparent, dark, blue-green or green colored plastic called celluloid.
The reason some people wear those green plastic visors while playing billiards is to shield the light from the eyes, reduce glare from low hanging and very bright pool table lights and to help see the billiard balls more naturally against the cloth.
I believe the reason for the visors being the color green; apparently it helps shield glare and blocks certain parts of the light so that the balls on the pool table can be perceived in a 'true and natural' manner.
Fast Larry (a famous and very experienced billiards player from Atlanta GA) has written about the use of these billiards visors in modern cue sports. To this day he wears a visor while playing pool. Below is an excerpt from a rant by Fast Larry:
Golfer wears hats to deflect the light from the glare of the sun from blinding their eyes; I do the same thing with my pool visor.
Overhead pool table lights are hung too low and usually have very bright incandescent lights that if you eye catches one it can affect your vision for some time. This constant glare is not good on the eyes either. [...] Try a visor to play in, you might like it. When I try to play with out mine the amount of glare coming in really affects me badly. Don’t be afraid to be the only one on your team or in your room to begin wearing a glove or visor. [...] Golfer wears hats to deflect the light from the glare of the sun from blinding their eyes; I do the same thing with my pool visor.
Many pool halls ban the use of hats and the employees do not understand that a billiards visor is more like a billiard accessory than a piece of clothing, and that a billiard visor is definitely not a hat (Fast Larry also argues this point very well). The VBSA (the Victorian Snooker & Billiards Association) of Australia has a dress code for all of it's member players and visitors to follow during representative play. It strictly prohibits wearing any type of hat, but it specifically calls out an exception allowing the use of a billiard or snooker visor during active play.
Hats are not acceptable items of clothing worn in RSL clubs. Visors are permitted whilst actively playing billiards or snooker.