All the sources I've perused can, just as Wikipedia does, only surmise on the how and why gunpowder made its way to Europe.
The Oxford Encyclopedia of Medieval Warfare and Military Technology offers a nutshell overview of the possible routes that might have been taken:
Just how the secret of gunpowder traveled west-ward to Europe will probably never be fully known, although it seems likely that there was not just one route but several—via the ancient trading route known as the Silk Road; by travelers from the west; by the Mongols; or by peoples of the Russian lands.
That said, a Dr. Guangqiu Xu in "China at War" provides a persuasive answer to both the how and the why:
When gunpowder's advantage as a weapon was made clear, the Chinese began to apply gunpowder to warfare. They started experimenting with gunpowder-filled tubes.
By the thirteenth century, Chinese military forces adopted gunpowder-based weapons technologies such as rockets, guns, and cannons, and explosives such as grenades and different types of bombs for use against the Mongols when they attempted to invade and breach the Great Wall on the northern borders of China. After the Mongols conquered China and founded the Yuan dynasty (1279–1368), they used the Chinese gunpowder-based weapons technology in their invasion of Japan and Korea, and other countries.
... at the beginning, the formula for making gunpowder was not common information, and only a few special weapon makers knew how to make it. This hazardous and highly explosive weapon, however, spread to Europe through the Silk Road, the world's oldest and most mysterious trade route.
In the tenth century, Arab scientists began to study and carry out experiments with gunpowder and its applications in warfare. When Europeans invaded Arabian countries during the crusade movement from the eleventh to the thirteenth centuries, the Arabs used their newfound weapon again the Christian troops, sparking both fear and interest from the crusaders. The technology would later be adapted by the Europeans for use in their military.
The Arab-Crusades theory is bolstered by the fact that gunpowder was initially known simply as black-powder or more suggestively, as Saracen powder:
Historians believe that gunpowder originated in China, where the 'black powder' or 'Saracen powder' had been used to manufacture explosive bombs and rockets since at least 1,000 BC. Knowledge of its powerful and destructive secrets most likely traveled west along the ancient Silk Road, a trade link that crossed the Asian continent and eventually linked the kingdoms of Europe with suppliers of exotic spices, gems, and of course silk. In the 10th century, Arab warlords had perfected the science of using gunpowder and turned its awesome power against the Crusaders, and this triggered the adoption of the black powder as a warfaring tool in the West.
Although the dates do not really conform to the Mongol defeat of the Chinese in an earlier excerpt, in "Firearms: A Global History to 1700", Kenneth Chase suggests that the Mongols were the ones who brought gunpowder westward:
The Mongols were probably responsible for bringing gunpowder and firearms to Europe. Chinggis Khan organized a unit of Chinese catapult specialists in 1214, and these men formed part of the first Mongol army to invade Transoxania in 1219. This was not too early for true firearms, and it was nearly two centuries after catapult-thrown gunpowder bombs had been added to the Chinese arsenal. Chinese siege equipment saw action in Transoxania in 1220 and in the north Caucasus in 1239–40.
William of Rubruck was a Franciscan friar who traveled to the court of the Mongol khaghan Möngke between 1253 and 1255. Although his account of his journey did not circulate widely in Europe, one person who took a keen interest in his experience was Roger Bacon, a fellow Franciscan. Whether by coincidence or not, the earliest European reference to gunpowder is found in Bacon's Epistola de secretis operibus artiis et naturae from 1267.
From "A Brief History Of Rocketry":
The rocket seems to have arrived in Europe around 1241 A.D. Contemporary accounts describe rocket-like weapons being used by the Mongols against Magyar forces at the battle of Sejo which preceded their capture of Buda (now known as Budapest) Dec. 25, 1241. Accounts also describe Mongol's use of a noxious smoke screen -- possibly the first instance of chemical warfare. Rockets appear in Arab literature in 1258 A.D., describing Mongol invaders' use of them on February 15 to capture the city of Baghdad. Quick to learn, the Arabs adopted the rocket into their own arms inventory and, during the Seventh Crusade, used them against the French Army of King Louis IX in 1268.
In summary, the question of why gunpowder made its way to Europe can very likely be blamed on its use in warfare. The how, on the other hand, is sketchy as, based on these sources, both the Mongols and the Arabs, or both, might have introduced it to the "Ferenghi" on the battlefield.