Some stories at the time just sort of... ended. Wikipedia said this episode was based on a short story called Disappearing Act (this is a book entry describing the story)
The gist of the story is that Bill is married to Mary, the latter of whom is forced to work a job because Bill is trying to start his career by writing. This makes both of them unhappy, but Bill is unwilling to make a career change that would at least make Mary happy by allowing her to stop working. He then strikes up a relationship with another woman who supports his writing. But his disconnect with the world causes it to start vanishing around him. First his girlfriend, then his friends, his wife, and finally the man himself are simply erased from existence. And that's how the story ends.
While this type of storytelling is unpopular today (Star Trek Voyager had a similar type of episode), a lot of these stories were originally written, which is a different medium from TV. Magazines were the popular culture of the day, and often stories were often vignettes meant to provoke the imagination more than tell a complete story. This helps make more sense of the Serling quote from Nathan's answer
“When Dick Matheson first wrote the story, it had nothing to do with astronauts. At least if I’m dealing with outer space, I can say something, someone [caused the disappearances], and I’ve got a little bit more going on.”
Remember that this episode aired in 1959. Sputnik was the relevant issue of the day and, with Yuri Gagarin more than a year away, coupling a story about people just disappearing with a story about astronauts probably made a lot of sense at the time, especially to TV executives.