A dangerous criminal is in the vacinity of Mayberry, and the State Police hardly think the local police are up to the task.
There are a number of things in this episode that are examples of a good story as well as a good comedy. The characters all act to their strengths, events from the first scene foreshadow the conclusion, and the story fits perfectly with the program's major themes.
Ken Lynch is a good choice for the Stae Police captain, he's dismissive and authoritarian at the same time and has good facial expressions to add to important moments. At the same time, the script does an excellent job setting up Barney and Andy as way too inexperienced to be of any help to the "invasion" of North Carolina State cops on the town. But, of course, Griffith gets the best lines as his homespun wisdom yet lack of overt ego wrap up the case in perfect order. Every line of the conclusion shows his character as humble and eager, and yet, more on top of the situation than the city slickers ever suspected.
There are lots of installments that emphasize this theme, but this is one of the few that transfer it to police work.