This is a great question! Sadly, I have not seen anyone address this topic. There are two ways we can attack this subject: the first is to consider what was going on during the 1950s and 1960s; the second is to discredit the premise of your question.

So, the first approach. This can only be a matter of speculation because there is no good way to scientifically link societal and cultural factors to the emergence of a group of legends. It is easy to imagine the rise of the Baby Boomer generation as a factor. The period's society (at least in the US) was being increasingly dominated by a large demographic bulge of young, sexually-charged peers in an environment with a lot of new technologies that allowed them to have an increasingly shared, homogeneous culture. Many of the legends in question dealt with young sexual urges, how they should be punished and how they were dangerous, and so it is easy to imagine that these legends emerged as a way to scare young people away from irresponsible, pre-marital sex.

That works to a point, except the poodle in the microwave - toasted by an older owner who thinks the new device will dry off her pet - doesn't have anything to do with that. With this, perhaps, we should look to the idea that our young generation enjoyed the horror, and just as there were a lot of horror movies appearing for date night, these legends offered a free, folklore counterpart for people who wanted to scare or be scared. But I'm not sure, and I don't believe anyone can really get beyond some imaginative suggestions.

Which brings us to our second point, namely that you are simply wrong! Legends (what folklorists define as stories told generally to be believed) have been around for millennia. We sometimes think of them as the property of the illiterate and of the pre-modern/pre-industrial world, but that idea is simply wrong. Literacy did not end the telling of legends. There is ample evidence of people telling stories across the barriers of literacy and industrialization. Some of these survived and were adapted for a modern environment: The Vanishing Hitchhiker, for example, has an earlier counterpart involving a wagon rather than a car. Many stories about extraterrestrials and UFOs sound a lot like stories about the Wild Hunt or about the fairies (who often abducted people, left circles in the landscape, and killed cattle).

I completed a study of a legend in the "Wild West" dealing with a stage coach driver: various writers picked it up, and as it began to show its age, Mark Twain demolished it, and turned it into a point of ridicule. With this, perhaps, we can see a hint at an answer to the inclination to see the 1950s/1960s as a fertile setting for the emergence of the urban legends. You are certainly correct in your perception, but that perception may be based on an illusion.

Given our Mark Twain example, I would postulate that with the emergence of an increasingly literate populace and with a lot of newspapers and increasingly other forms of mass media, legends often had a short life expectancy. I suspect many of us have heard someone discount a story with the declaration, "Oh, that's just an urban legend" - meaning it is a story that is not to be believed - and therefore should be killed! Once we take believe away, it is like taking the air out of a balloon. Nothing is left.

Perhaps what we are really seeing is not a new phenomenon, but rather the newest generation of legends, now for the first time labeled "urban legends" and with the demographic bulge of the baby boomers, there was a large peer group ready to spread the narratives and, in their egocentric conceit, to declare these legends as a completely new phenomenon - for the new generation (or as it was told by the Mad Men, for the Pepsi Generation).

I suspect that we are looking at a relatively new process or creation, spread, and disposal of legends, a process that was well in play by the nineteenth century, but that became particularly noticeable after WWII and with the rise of the Boomers.