How did the modern genre of fantasy become so heavily associated with the medieval period?

Contemporary fantasy settings in the Dungeons & Dragons mold generally falls under the "quasi-medieval" mindset. The quasi is important, because very few of these settings attempt strict historical accuracy, and a lot of the individual elements are often drawn from different periods of history...but there are some common elements. Gunpowder is usually absent, or at least not prominent; so combat is focused on melee weapons like swords and spears, and ranged weapons such as bows and crossbows. The feudal system is typically present, with a defined system of ranks of nobility and vassals. Industry and commercialism is usually limited, non-white Europeans are often absent, religious conflict rarely touched upon.

Which is to say, it doesn't look a lot like the period from late antiquity (c.5th century CE) to the early modern period (c.15th century CE) at all, which was generally a mess and marked by technological innovation, international trade, contact with other cultures (sometimes quite violent contact, but there you go), religious upheaval. But it bears obvious similarities to the courtly depiction of what the Middle Ages should have been like in chivalric romances like Le Morte d'Arthur, Sir Gawain and the Green Knight; chansons de geste such as The Song of Roland and epics such as Beowulf, the Nibelunglied, and the Eddas.

If you look at what was fantasy literature like before Tolkien, you can see this kind of approach as a mode in fantasy literature of the late 19th/early 20th century. As to why this quasi-medievalism appealed, I'm going to recycle an earlier answer:

I'm going to borrow from an expand on a previous answer a bit:

At the turn of the century, folks in the United States and United Kingdom expressed an interest in medievalism; it was a response to a number of different factors among all walks of life - increased literacy and publication of classics like L'Morte d'Arthur and La Chanson de Roland, medieval fairy tales aimed at children, class-based responses against increasing urbanization and industrialization of modern life, nouveau riche infatuation with symbols of traditional authority, military innovation and propaganda connected with World War I...

William Morris of the Arts and Crafts movement is a good starting point in many ways, because he was an important early figure in turn-of-the-century fantasy with his novels The Well at the World's End (1896), and his anti-industrial attitudes, the looking-back-on a medievalist setting with sort of rose-colored goggles had its echo in the works of Lord Dunsany, especially proto-Sword & Sorcery tales like "The Fortress Unvanquishable Save for Sacnoth" in The Sword of Welleran and Other Stories (1908) and eventually J. R. R. Tolkien's The Hobbit (1937) and The Lord of the Rings (1954-1955) - there are all authors who deliberately emphasized and ostracized the creep of industrialization, urbanization, and "modern life," praising the simpler ideals and slower life of rural English life (and in so doing, all set some precedents for early environmentalism); corresponding to this slightly were authors like Arthur Machen, who borrowed aspects of medieval legend and romance in stories like "The White People" (1904) and "The Bowmen" (1914).

But you have to contrast this "Romanticist" movement with works like Mark Twain's A Connecticut Yankee in King Arthur's Court (1889) and T. H. White's The Once And Future King (1958); these were no less fantastical, but much cynical (or if you like, humanist) in their portrayal of medieval life and society as something other than the "knight in shining armor" trope. And this reflected, to a degree, the real-life interest in the physical reality of medievalism.

In Chris Bishop's Medievalist Comics in the American Century, he looks at the theme of "medievalist" comics - taking as particular examples of close study Prince Valiant, Green Arrow, Conan the Barbarian, etc. as American comics seized on these "medieval" archetypes of knights, archers, and swordsmen as heroes in the 20th century, and on page 29 he notes:

By the time [William Randolph] Hearst had acquired the rights to publish Prince Valiant, he could already lay claim to a sizable collection of medieval objects. Hearst reputedly possessed the world's largest private collection of medieval armor, far outstripping his nearest rivals the industrialist Henry Ford and the banker Andrew Mellon.

Hearst was not interested in medievalism in the same way that Morris was - he was not casting back to a more romantic age as a refutation of industrialism, as much as he was enjoying the trappings of a traditional system whose elite status he had intruded upon with his new wealth. But it was a fascination with the trappings of the period that was shared across many different classes. For example, in Bashford Dean's Helmets and Body Armor in Modern Warfare - Dean was associated with the Metropolitan Museum of Art in New York City, and was a noted collector and cataloguer of arms and armor; his book, published during WWI, shows clearly how contemporary minds were struggling to come to terms with the change in technology and tactics - trench warfare really being a throwback to siege warfare, swords rapidly obsolete as side-arms but still present (as were bayonets), and in this book Dean focuses on the design and effectiveness of personal body armor, helmets, and shields, noting their forebears in medieval forms. There, Dean noted that Ford was a notable manufacturer of helmets for World War I, and writes on page 212:

Thus, in the summer of 1917, this helmet, after having been shown to the experts of several manufacturing concerns, who feared that it could not be made, was turned over to the Worcester Pressed Steel Company with directions to produce it in an experimental lot in 12 per cent manganese steel. Dies accordingly were prepared and every effort was made to deliver the helmets needed. Unfortunately, however, the dies which this firm employed were inadequately designed and in the end the only helmets produced were defective, having their sides wrinkled and their tops thinned out. Later, however, this helmet was shown to Messrs. Ford and Company of Detroit who declared that it could be pressed and pressed well without an important breakage of metal. Accordingly, this firm, received an order from the Ordnance Department, produced a set of experimental dies. On these, during the fall of 1918, a couple of thousand helmets were turned out. It may be mentioned that the principle upon which Messrs Ford and Company proceeded was quite similar to that which an armorer would have used in olden times, for the top of the helmet was pressed not as a final but as an earlier operation.