The term rewrite man is something of a misnomer. Some are women, of course, and rewrite men do not just "rewrite". They take notes gathered by on-the-scene-reporters, information gathered by telephone, or from wire services or clippings from other newspapers, and write articles.
Sometimes an entire front page, with bylines from several different reporters, will have actually been written by a single rewrite man working with an editor.
The job has lost much of its importance due to technology that allows reporters to write and transmit articles from the field. In the pre-computer days of newspaper work, however, it was vital. At the most extreme example, reporters on deadline would telephone into the newsroom and dictate their notes to an editor - hence the movie cliché of reporters rushing to telephone booths and shouting "Get me rewrite!" into the phone.
Rewrite men are common at large national magazines, where reporters handle the reporting of the story, while a writer takes the material and writes the actual article. In this case, the reporter will put all the information and quotes gathered into a multipage file which is then given to the writer for writing. In cases of a story in several areas, several reporters will write and give their files to the writer. The article which is written has all of the information and observations by the reporter(s), but the writer never saw this. People Magazine stories are written in this format, with field reporters doing the reporting and New York based writers writing the pieces.