Before a film or TV show gets to the production stage, someone has to "vet" the script and assure the producers that it's a production worth making. This is the "development" phase of a project, and the job of vetting the script often falls to a script reader or story analyst who knows what elements make a good production. That's why script readers often hail from the screenwriter ranks -- or at least have some type of production background.
Gain Experience and Make Connections
Script readers typically start by either studying screenwriting or film, or they work their way up the ranks in the film and television industry. If you haven't studied film or screenwriting in college, seek out entry-level jobs as a production assistant or a development assistant on websites such as Mandy.com, Craigslist or Backstage. Do good work and establish relationships with the development staff in the production company, offering to do "coverage" for scripts on a volunteer basis. If you're already working as a screenwriter, joining the Motion Picture Editors Guild can help you make connections -- and earn more money -- as a union "story analyst." Contact development directors at studios you've worked with, offering to do coverage on scripts for low pay, or even volunteer to gain experience. Also seek out organizers of film festivals or screenwriting contests. As you gain experience, your connections can lead to steady work.
Honing the Craft
Doing coverage involves writing a summary of the script and offering your insight into whether the show or movie is worth making. To pick the good scripts from the bad, spend plenty of time honing your craft. Read other people's coverage of scripts to gain knowledge into what is expected from a script reader. Also read as many screenplays and teleplays you can get your hands on to get familiar with the format, and ask people to review samples of coverage you've done to get feedback. Websites such as Simply Scripts or Scriptologist offer sample scripts and even samples of coverage to get you started.