Continuity mistake: At the start, when Andy stops at Wally's station Gomer is telling Jason about his Trout bait, then someone pulls up behind Jason's car and they're bumper to bumper in the wideshot, but in the exterior shot facing Jason there's no car behind him, but that car is back again before he pulls away.
Continuity mistake: When Opie shows Andy his arm with the chore list written on it, Andy uses his right hand to hold up Opie's arm, but in the closeup it's Andy's left hand holding up Opie's arm. (00:03:50)
Continuity mistake: In earlier episodes, we are told that Gomer can't do mechanic work. He just pumps gas and air: "There's no charge for air; air is free; we don't make no charge for air." But in this episode, Gomer has become an ace master mechanic.
Answer: As noted in the previous answers, in real life, things like this provided wind and/or rain deflection, and also maintained a bit of privacy when blinds were raised somewhat. The interior courthouse set was located in the studio, so the "outside" Main Street didn't exist. I believe these things were added to the courthouse windows for practicality, to avoid some crew movement being visible on the opposite side of those windows. These are not "window boxes" to hold anything, as they're actually bottomless; we can see the Venetian blind's long pull cords under them. They're made of plywood and simple to build, so the "material and labor" was inexpensive. Similar variations made of different materials are in other movies/shows. In 1957's "12 Angry Men," textured chicken wire glass panels are in the jury room windows, and in "Jesse Stone: Night Passage" another type is in Jesse's office windows.
Super Grover ★