Looks to me like a "bump stop" inside of a torn or cut dust boot? Normally the little rubber sleeves are the "dust boot" that acts like a shield that helps keeps water and dirt from entering the shaft area. Either way it looks to be old and crusty from the pictures, most newer shocks may not come with dust boots or bump stops for that matter, but the dust covers do help keep the weather elements out of the shaft seal ,which is what you want, I do not know any one who would not want to prolong the life of their shock absorber.
Will they work without them? OF COURSE, but with sacrificed longevity. I personally use them on ALL my daily drivers, the thought of dirt and water possibly pushing past a shaft seal after repeated abuse and use does not entertain me in a positive manner, not for the kind of money we all spend on these cars, we are talking a extra 20-25 bucks on a suspension that cost hundreds to begin with.
Now that I think about it, even my Fortune Auto Coilovers came with them and those go on track cars for the most part, so obviously there is a benefit or all these engineering companies would not be using them on their cars and on their performance products which lay down record lap times.