Why did the old waiter want a "clean, well-lighted" place?
I assume you're talking about the end of the story when the older waiter from the cafe goes to the bar before going home. The older waiter is disturbed about what happened in the cafe with the old man and the younger waiter because he identifies with the old man. He too is lonely with no one to go home to. He can understand why the old man would want to drink at a cafe rather than a bar. The older waiter, not ready to go home and face his loneliness and insomnia, goes into a bar. He immediately notices that it's well-lighted, but it isn't clean. He realizes that "a clean, well-lighted cafe was a very different thing", and he could identify with the old man taking pleasure in sitting in the cafe rather than a bar.
A clean, well-lighted place holds at bay the "nada" that he perceives in life: "What did he fear? It was not fear or dread. It was a noting that he knew too well. It was all a nothing and a man was nothing too." "Nothingness" or "nada," which he repeats over ten times, goes beyond loneliness, consisting of a sense of meaningless in the universe akin to existential angst--a sense that there is no reason for anything,that there is no purpose or meaning to life. Loneliness is the immediate cause of this, perhaps, and certainly living in the "light" would mitigate the discomfort of being in the "dark," where one in fact can see nothing.