All you had to do was read your cue cards as fast as you can.
When you watch old episodes of Dragnet and see those actors standing there reading their teleprompters as fast as they can, you feel like maybe YOU could be an actor, too!
The show went from radio to television in the 1950s. Unlike most shows that made this switch, it kept the same cast.
On the radio, actors don't memorize lines. They stand in front of the microphone reading their script. Webb preferred this, so when the show went to television, pretty much all the dialog was on cue cards and teleprompters. Actors rarely had to remember lines.
If you watch the show, at least the color episodes, you can see that all the dialog scenes are shot in a studio. There is an establishing shot with a voice over narration, then the dialog is done in a studio in front of a rear screen projection. It worked pretty well most of the time. Could be done fairly easily now in front of a green screen.
It seems like it would be a good approach. You could get better performances both from stage actors who tend to over do it, and non-actors who usually talk very, very slow.