Barnie lets his head get turned when a pretty new resident of Mayberry asks him for directions to the post office. As a result, he forms a two car motorcade complete with siren and leads her car across the street. Could he not have just pointed? What's worse, Thelma Lou saw the entire thing.
This leads to an argument and break-up between Barney and Thelma Lou, and Barney ends up rebounding and dating the new girl whose presence started the argument in the first place. But then one night the new girl misunderstands Barney and believes he's proposed marriage. And when he tries to back out of the situation, she threatens to sue him for breach of promise.
Barney always was a bit of a lady's man on the show, and this time his instincts may have gotten the best of him. As usual, it's Don Knotts's nervous reaction to the situation - his facial expressions, his nervous pacing, the rising pitch of his voice - that is priceless.
States started repealing breach of promise laws starting in the 1930s, but North Carolina did not repeal its law until 2019. So Barney could have still been on the hook in 1961. These laws were usually enforced by women against men due to the fact that, at one time, a promise to marry was pretty much a promise to support.