I want to note that it is not just an unsourced claim that you'll see among people online seeking easy narratives, but is indeed a Known Fact among fashion historians, based on what you see in illustrations of the period, which is why you tend to see it stated without sourcing.

Finding the "origin" of a fashion is much more difficult than you'd think, because fashion is never simple - it's rarely or never the case that someone influential accidentally makes a change and others copy it, or that some new technology comes along that suddenly makes people relate differently to their clothes. (There are certainly ways that technology relates to fashion, it's just rarely so clear-cut.)

I’m not aware of many scholars looking into the actual matter of why people chose to start tailoring - in large part because much of medieval fashion history focuses either on artwork or on a more literary/psychological angle - outside of one source: Odile Blanc's "From Battlefield to Court: The Invention of Fashion in the Fourteenth Century" in The New Middle Ages: Encountering Medieval Textiles and Dress (2002).

It’s the 1340s when chroniclers, always ready to moralize about indecency in dress, started to discuss the issue of extravagantly fitted clothing (first in men, later in women), as well as extravagantly expensive fabrics. Not only was this clothing shockingly tight, it wasted fabric both by excessive cutting (to shape the clothing to the body, and to create dagged edges) and by having things like loose, hanging ends on sleeves. Fabulous embroidery, jeweled belts, and flashy hats also came into style. All of this seems to have appeared in major European cities around the same time, rather than clearly starting in one place and spreading – although moralists of different countries blamed each other for infecting them with the new style. This was very much related by them to the onset of the Black Death, which was God's punishment for such foolishness.

This was also the beginning of the Hundred Years War, set off in the late 1330s by Edward III. As I explained in this previous answer on Henry VII,

Henry V's claim to the French throne went back to Edward III (father of John of Gaunt; his own great-grandfather), whose mother was Isabella of France, daughter of Philippe IV. When Charles IV died, Edward was willing to go along with the idea that women couldn't directly inherit, but felt that he had the greater right to the throne as the most closely related man in the family. Philippe VI was a more distant cousin of Philippe V and Charles IV, but it was declared that women couldn't even pass on a claim, and he was a male-line relation and, probably crucially, was able to act on it and take it immediately (much like Stephen of Blois usurping Empress Matilda). And Henry was not reaching back many generations - Edward had pressed this claim, and his sons would continue to as well. It was the central issue of the Hundred Years' War!

In this time of war, more and more men - including aristocratic men serving in command posts on the battlefield - were getting involved with military dress, a central piece of which was the pourpoint. This was a quilted/padded garment worn underneath or in place of the armor. See, the issue isn't that they had no clue how to cut clothing in ways other than a T-tunic - it is simply that that wasn't what they generally wanted clothing to look like. But with lots of men wearing pourpoints, and the garments needing to be made to fit rather closely to fit properly under armor, they caught on as a more general fashion. The body-shaping allowed by this tight and stiffened garment (when modified beyond what was required for fitting under armor) became widely interesting to people across the continent, men and women, and a new fashion was born.

Before I answer, a gentle correction: The changes in tailoring you're referring to date to the mid-14th century, which is the 1300s.
That aside, art historians have long noticed the changeover that happens in the mid-14th century to first men's, and then women's high-status clothing, and noted the similarity in the men's clothing to padded garments meant to be worn underneath armour (often known as pourpoints). In particular, the requirement for a close-fitting garment to also provide the ability to move one's arms led to changes in the way sleeves were fitted, pivoting away from the straight-cut sleeve head to curved heads, culminating in the extreme example often known as the "grand assiette" sleeve. Knowledge of a few extant examples of these garments (such as the pourpoint of Charles de Blois) give credence to this theory, probably best presented in Stella Mary Newton's classic Fashion in the Age of the Black Prince. It's usually thought that these fashions got their start in Italy and moved northwards. The changes in cut did lead to changes in tailoring in general for high-status clothing and eventually, by the 16th century, innovations such as patterning books and the sense of various courts being seen as "fashion leaders" (for instance, Spanish fashion being prominent in the 16th century, as the discovery of new dyestuffs in the Americas led to better ways to dye black, allowing it to supplant red as the colour signifying wealth).

With these changes well documented both in artwork and in literature, early 20th century fashion historians such as Boucher and Laver definitely seized on it to posit that the changes in the 14th century signified the emergence of "fashion." Often a popular reason (aside from the influence of military fashion) given for the changes was that the ravages of the Black Death led to a love for the extreme (and, indeed, some contemporary chroniclers do point at the "extremes" of male clothing as an example of the kind of depravity that arose in the wake of the Plague. ) However, none of the changes of the 14th century are sudden. For instance, you can start to trace the close fit of sleeves starting in the 13th century for both men and women, even as the bodies of garments remained loose. For men, the length of tunics was also starting to shorten, and hosen were known for both sexes dating back even farther.

More recently, Sarah-Grace Heller, in Fashion in Medieval France, has found significant support for the fact that the idea of "fashion" was not necessarily born in the 14th century, but can be found in literature as far back as the 11th century. and was becoming well established in France by the 12th century and flowered in the 13th century. In other words, the tailoring changes did not create "fashion as we know it", but reflected the growing importance of clothing and dress in noble society--a process that had been developing for over three hundred years. Another significant factor was the development of the cloth trade--both high-quality wools as well as increasing access to silks.. Heller makes a good case for the overlooked 13th century as being the century where the idea of "fashion" begins to mature, especially in the use of fabrics. The 13th century featured long, fairly simple, draped styles for high-class women (in particular), evolving out of the increased use of silks in the 12th century (the century associated with the tightly-laced, but also flowing bliaut styles).

I really recommend Heller's chapter "The Seduction of the Well-Dressed Form" if you'd like to get an overview of how the idea that nothing before the 14th century was "fashion" took root with modern costume historians.