Tolkien's works were borderline transgressive when they were released. I don't have the time to dig into a lot of the details at the moment, but consider a few things:

First, the reasons for popularity. They're good stories with interesting characters, many complex themes, and interesting settings and events. There's a lot of the classic "hero's journey" elements and mythological elements, those stories have always been popular. Tolkien studied the Norse sagas and epics so a lot of those materials and that form of story telling ended up in his books.

Second, the romanticism. Tolkien's works harken back to the 19th century romantic period, with a clear favoritism shown for nature and peacefulness over industry and violence. This was out of fashion by the mid 20th century, and even looked down upon. Romantics were viewed as empty headed and fantasy fiction was viewed as escapism. Remember that the society into which Tolkien's works were released was the same one that the counter culture movement rebelled against. In Tolkien's world you have people such as the Hobbits who are not especially industrious and in fact could be regarded as lazy. As Tolkien put it they "valued food and cheer and song above hoarded gold", and he lauded them for it fairly explicitly in his books. That very much ran counter to the prevailing zeitgeist in the developed world of the time. But it resonated strongly with the counter culture movement which included elements such as "hippyism" which shared that sentiment.

Third, the religious aspects. The mid-20th century in the US and even in Europe was still an era where religion had a supremely powerful place in culture and society. On its face, Tolkien's works are strictly blasphemous against Judaism, Christianity, and Islam. They describe a world where there is no God of the bible, a world with a different god, a different creation mythos, a different people, and so on. This is more transgressive than it may seem because it's not just a rejection of, say, Christian orthodoxy along the lines of "hey, here's this world and it doesn't even have God in it, and it's not a big deal that it doesn't". More than that, because it rests on its own creation mythos, its own deep historical narrative (which religions had traditionally taken on as one of their "core competencies"), its own world, its own laws of nature, it serves as a tool for intellectually deconstructing established religions. One can look at the toy mythology of The Lord of the Rings and easily make comparisons to, say, Christianity, and in so doing come to the realization that Christian mythos could just as easily be the work of man rather than divine revealed truth.

Again, the counter culture movement of the '60s and '70s were also on the track of deconstructing institutions that relied heavily on dogma and orthodoxy, especially religions, so these ideas resonated strongly with them.

Embracing Tolkien's works became not just about the quality of the story, it became a political statement. A rejection of the insistence of "the man" to buckle down, get a boring job, put away toys and escapism, and become a cog in the great socio-economic machine of the times. And a rejection of the power structures that supported (or were perceived to support) petty warfare, inequality, greed, industrialization at the cost of nature, religious doctrine, and so on. During the '60s and '70s graffiti or personal paraphernalia (such as buttons) proclaiming "Frodo Lives!" served as one of many shibboleths for the hippie/counter culture movement. Those simple words were seen as a stirring rejection of establishment ideals and power.

And fourth, the detail. Tolkien's world had a depth and complexity that was far beyond the average story. Part of the lure of the stories is their setting, a fully fleshed out world with its own mythos and thousands of years of civilizational history. Tolkien created a deep backstory which creates a palpable richness around every element of the tale. Gandalf's sword, for example, is not just any sword, it's a sword with a name and a six millenia history stretching back to the war against Morgoth (who used to be Sauron's master) when it was wielded by a legendary Elven king. Similarly, Frodo's Phial given to him by Galadriel was filled with water from a fountain that held the light of Earendil's star, a story of incredible depth and complexity which Frodo and Sam allude to in various comments from time to time. And that sort of depth is indicative of the story as a whole. Everything and everyone everywhere has a history, it makes the world feel more interesting and more real, providing a degree of verisimilitude to the characters, peoples, and places. That depth and breadth enabled an entire realm of conversation about Tolkien's works beyond merely the story and its meaning. One could discuss the various races or peoples of the books and delve into the details or minutiae of them. One thing that sets apart great literature from merely good literature is that great literature is typically not just one story tied up in a tidy package, it's a whole gamut of ideas, concepts, events, and creations which serve as the kernel of potentially very open ended discussions and thoughts. In Tolkien's works there were a great many such kernels. As such, it inspired a great many people to tell their own stories using similar themes, elements, or settings. Dungeons & Dragons, for example, was set firmly in a Tolkien-inspired world. A world populated by dragons, elves, dwarves, wizards, hobbits (later renamed halflings), and so on. A world where a typical adventure involves a story line that very closely mirrors either The Hobbit or The Lord of the Rings (a diverse band of adventurers goes on a grand trip, crawls through ancient caverns and dungeons, fights goblins, trolls, dragons, and evil wizards, and so on).

Tolkien's world, characters, story, and themes resonated with a group of people in a way that other contemporary literature did not. And the depth of detail made it a suitable kernel for a great many derivative works as well as helped maintain a momentum of enthusiasm in the work. It also served as a calling card for the counter-culture movement which helped to increase its profile and popularity.

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