Why did it happen in the 14th century, when the blast furnace was invented much earlier? Can someone explain, how medieval blacksmiths figured it out? What discovery enabled them to create plate armors?
The emergence of late medieval full plate armour wasn't really prompted by any specific discovery or advancement in metallurgical tech. Partial plate armour, in principle, can be traced all the way back to Classical Antiquity, such as the Greek muscle cuirass later Roman lorica segmentata.
Rather, the most critical development was the appearance of larger bloomeries. It was no coincidence that plate armour began in North Italy shortly after such bloomeries appeared there - they made it possible to produce sufficiently large steel plates in one piece.
The suit of armour of large articulated plates first appears in 14th century Italy, followed later by Germany and the rest of Europe. This was a result of bloomeries having grown large enough to produce metal plates of the required size.
Williams, Alan. "The Metallurgy of Medieval European Armour." Proceedings of the Forum of the 4th International Conference on the Beginning of the Use of Metals and Alloys. Shimane, Japan, 1996.
The problem before this was that you need quite a sizeable amount of steel, about 10kg, for a breastplate. Prior to the late 13th century or so, European bloomeries were generally not large enough to produce so much steel in a single chunk. To make a steel breastplate then, you'd have to weld two or more separate plates together, which compromised its protective value despite an enormous price tag.
A plate of armour which weighs between 2.5 and 4.5 kg will pose new problems lo the producers. Billets of metal of 10kg or more may be needed to make such a plate
Williams, Alan R. The Knight and the Blast Furnace: A History of the Metallurgy of Armour in the Middle Ages & the Early Modern Period. Brill, 2003.
A significant factor behind the rise of larger bloomeries was that Late Medieval Europe began harnessing the power of rivers - using waterwheels to power furnace bellows enabled larger blooms of steel to be produced.
Once decent full plate armour became feasible to create, the main obstacle to adoption - aside from a lack of need early on - was simple economics. A full set of plate armour was extremely labour intensive to forge, and consequently very expensive. Keep in mind, even the Romans had to abandon the lorica segmentata after the Crisis of the Third Century rendered it economically and logistically unsustainable. No military in medieval Europe could rival the Roman Empire's resources.
By the 14th century, however, smiths had begun using watermills to driver hammers for shaping the steel, greatly reduced the labour required.
Water-power enabled smiths to increase their output. Bellows driven by a waterwheel could produce a continuous powerful draught from a free energy source, so it was at last possible to enlarge the size of the furnace and the bloom thus produced. Water-powered hammers were also heavy enough to fashion the larger blooms.
Blair, John, W. John Blair, and Nigel Ramsay, eds. English Medieval Industries: Craftsmen, Techniques, Products. A&C Black, 1991.
It was the newfound relative affordability of plate armour, combined with improved designs reducing its tactical downsides, that ultimately enabled its adoption. The full plate armour reached its peak about the same time advancements in projectile weaponry began to render it obsolete, however.