There's no Rod Sterling voice as you heard the phrase, "...you've entered the Twilight Zone" or something similar. All had to do was set back and enjoy the ride. Each episode completely unique onto itself which incorporated either suspense, horror, sci fi, or fantasy.
The original 156 episodes aired from 1959-1964. The show hired some of the top talent of the day. Writers included Harlin Ellison, Ray Bradbury, Jerry Stohl, Richard Matheson, and Rod Sterling himself (who is vastly under credited as a writer).
The Twilight Zone has been part of pop culture since it's original airing. It has spun off a movie, a re-make of the series, a kick ass pinball game (man I wasted a lot of quarters on that one), and comics. And there's no way to calculate the number of people it has influenced over the years from writers, directors, musicians, and many others.
If you've never witnessed the greatness, do yourself a favor and start watching.