This is a fairly broad question, both because of the breadth of the time period we know of as "medieval" and because of the long amount of time since the end of that time period (around 500 years). However, I can offer one piece of the puzzle:

In the United States (many of the trends I will describe also apply to England, but I'm an expert on the U.S., and there are enough differences between the two situations that I want to tread carefully) towards the end of the nineteenth century there arose a new phenomenon among the cultural and intellectual elites of the country. Those elites had once also been the financial elites, i.e. the old upper classes of the Northeast (the Boston Brahmins are the most prominent of these), but they were rapidly being eclipsed in financial power by newer industrial capitalists. Anyway, these Northeastern cultural elites, exemplified by figures like the historian and diplomat Henry Adams and his friend the painter John La Farge, began to feel a new sense of dis-ease with the conditions of modern life. This condition, which was characterized by physical and psychological symptoms including fatigue, lassitude, and a general sense of disquiet, was called neurasthenia. The historian T. J. Jackson Lears, on whose No Place of Grace: Antimodernism and the Transformation of American Culture much of this post will be based, identifies industrial capitalism as the root cause of neurasthenia. Specifically, industrialization and capitalism had reached a kind of saturation point by around 1880, whereby economic relations were now entirely bound up in the impersonal sphere of the market. This process obviously caused the most changes in the lives of more ordinary people, the workers who were staffing these increasingly large and dangerous factories. But it also affected the upper classes, particularly those whose economic stature was in decline in comparison with the new moneyed elites, your Rockefellers, Morgans, etc.

One of the primary cures for neurasthenia, and for the more broadly defined feeling of malaise and banality that affected many people of this time, was to seek out new experiences, particularly exotic ones. For many, this meant traveling in Asia and Africa. La Farge and Adams, for example, traveled to the South Pacific and Japan in search of a change in perspective. The French painter Paul Gauguin famously sought primitivism in Brittany and then later in Tahiti. Another was to valorize the nobility of the independent artisan, so recently rendered extinct by industrialization. This saw its most complete manifestation in the Arts and Crafts Movement, which got its start in Britain but soon made its way to the United States. William Morris, the British founder of the movement, was a socialist, and saw in the medieval guild system a way to return the means of production to the laborers. He and his followers organized groups of artisans to create decorative and functional objects that could be sold directly by producers to consumers, thereby keeping the profits of their labor away from the capitalists. As an economic movement the Arts and Crafts had shall we say limited success, though Arts and Crafts artisans and artists created a wide variety of very beautiful and sought after objects, as well as contributing to the development of Art Nouveau, and thus to artistic Modernism more generally.

The figure of the artisan was not the only way the medieval was valorized at this time. Americans (and their English Protestant counterparts) also began to be interested in some of the other cultural trappings of Medieval Europe, most notably Catholicism and what we might think of as medieval adventure (knights and whatnot). They were attracted to Catholicism, which was regarded with such a high degree of suspicion in the United States, for what they saw as its mystical qualities, the culture of miracles, saints, and all that sorts of stuff. In comparison with straightforward Boston Unitarianism, medieval Catholicism and its rituals, rigid hierarchies, and material splendor was quite attractive. Similarly, Arthurian legend and other tales of adventure, daring, romance, etc. were incredibly appealing to people who felt stifled, not just spiritually but physically, by the courses their lives were taking.

So, in one sense American interest in the medieval comes at the very moment when the last vestiges of the medieval economic system were finally being replaced with industrial capitalism. This interest was quite specific, and it stemmed from a desire to experience the exotic. It was accompanied by an interest in Buddhism, African art, and so-called primitivism more generally. It also sprang into being at the exact moment of the birth of high imperialism, when Europe and the United States began to scramble for the bits of the world they didn't already control (and when the United States was finally consolidating its centuries-long conquest of North America's original inhabitants, as well as the end of Reconstruction and the establishment of racial apartheid in the US). Thus, we might more cynically see it as an attempt to turn away from the horrors unleashed by capitalism and conquest. In any case, I'm honestly not sure what relationship this turn-of-the-century interest in medievalism-as-exoticism has with later literary and pop culture interest, and I'd love for someone to be able to connect the two.

The notion of the “Middle Ages” developed during the Renaissance as a somewhat contemptuous term – Renaissance humanists saw ancient Rome and Greece as their progenitors, and devised the term “medieval” to refer to a prolonged period in which European lost sight of Classical ideals.

This perspective was also adopted by Enlightenment philosophes, who viewed it as age of ignorance, superstition and barbarity. Molière famously referred to Gothic architecture as “hateful monstrosities vomited up in torrents by barbarians throughout the centuries of ignorance,” while Voltaire called the Middle Ages “a barbarous history of barbarous peoples, who became Christians but did not become better because of it.”

Voltaire and other eighteenth-century thinkers felt that science and rational thought had been abandoned during the Middle Ages, and consequently associated the entire period with a plodding religiosity and superstitious ignorance. They also were repulsed by the medieval political order, which they viewed as being based on violence, coercion and mindless tradition.

Even the dashing valour of medieval knights was subject to criticism; Cervantes’ novel Don Quixote is a mocking satire the chivalric codes of the knightly class.

Yet there was also a countervailing tendency toward Romanticism in the 19th century that held the Middle Ages in higher regard. This valorization of the medieval was expressed through the collection of folk songs and folk tales (e.g., the Brothers Grimm), or the paintings of the German Nazarenes.

But in the English-speaking world, no one did more to rehabilitate the image of the Middle Ages than Sir Walter Scott. An avid collector of folk tales and student of medieval history, he wrote several bestselling novels that reshaped popular imaginings of the Middle Ages. In Ivanhoe (1819) and Quentin Durard (1823), he evoked a world of chivalric romance, with daring knights and maidens in distress. The popular conceptions we have today of the Middle Ages – such as the “knight in shining armour” – mostly derive from Scott’s novels.

Scott’s work was popular not only popular in Britain, but also became all the rage in the U.S. South, and many Southerners began to think of themselves as the inheritors of a chivalric “Cavalier” tradition that had its origins in Arthurian tradition. (The Ku Klux Klan, for example, evoked this imagery to a very great degree.)

Sources: Robert Bartlett, Medieval Panorama; Alice Chandler, “Sir Walter Scott and the Medieval Revival”