though Wikipedia details the changes in armor reasonably enough, i can't find any information about the advancements in smithing and metallurgy that allowed those improvements, and whether at all they were the result of such advancements or just cultural changes, necessity, or a better armor construction. sorry if it's a bit broad.
This is an important question in the history of armour. In fact, this is a huge question, and there's a lot to say about it.
I will start off with some historiography. The early historiography of armour is saw medieval armour's development as being like that of early 20th century battleships, or 20th century tanks - more powerful weapons necessitated better armour. In this telling, plate armour is a response to the crossbow and the longbow and to various staff weapons. Speaking abstractly, this is a demand-side explanation for the development of plate armour - plate armour develops because soldiers need it. But the chronology of this development does not quite make sense - crossbows appear in the 12th century or earlier, and plate armour does not really take off until the 14th century (after getting its start in the 13th). Moreover, some attacks, like a blow from a mounted soldier's couched lance, are at least as strong as bow-shots, and couched lances have been around for centuries before plate armour is developed. Now, some armour (especially later in the history of plate armour) appear to be responses to more powerful weapons, so we should not discount this demand-side explanation, but it is not the whole story.
Alan Williams in The Knight and the Blast Furnace looked at the bloomeries and blast furnaces that produced medieval iron and steel to provide a supply-side explanation for plate armour's origins. He looked at the size of excavated blooms and found that they greatly increase in the later Middle ages. This is important because armour cannot be welded from multiple pieces-each plate of a set of plate armour must be made from a single piece of steel (otherwise the armour is weak where it is welded). Big plates like breastplates must be made from big pieces of steel. This is a contrast to mail or scale armour, which can be made from small pieces of steel or iron (hence why societies with even small bloomeries can make these types of armour).
This all means that armours made of large (ish) solid places of metal need to start with large blooms of iron or steel in the bloomery. And there is a strong correlation between the increase in bloom size and the size of armour plates - from the introduction of one-piece breastplates to the profusion of single-piece helmet skulls, over the course of the 14th and into the 15th century the general trend is for the big plates of armour to get bigger. So bloom size (which is in turn the result of larger bloomeries) is an important precondition for the development of plate armour.
The other thing that bigger, more efficient bloomeries and blast furnaces do is make steel and iron cheaper. This causes a profusion of plate armour in various forms across different classes of soldier, until in the 16th century mass-produced 'munition armour' is arming the soldiers of the Early Modern monarchs. Blast furnaces which melt iron and produce pig iron (an iron/carbon alloy with a carbon content higher than steel) are cheaper and more efficient than bloomeries, which further reduces costs. However, pig iron must be 'fined' in order to be forged - it must have its carbon content reduced in order to make it less brittle. This process generally reduced armour's carbon content to little or nothing, which reduced the quality of the armour. This is particularly important because only medium and high-carbon steel can be hardened by heat treatment - 'quenching' hot steel in water to change its crystalline structure. So in the 16th century armour gets cheaper, but in some ways its metallurgy declines.
There are other technological developments important in the history of armour in the later middle ages. Water-powered trip hammers were used to beat blooms into sheets of steel, which would then be sold to armourers both locally and abroad. Water-powered polishing mills reduced the labor required to polish armour, which was the most laborious and tedious part of making armour. Both of these developments help encourage the spread of plate armour, or at least lowered its cost. But these cannot be taken outside of the larger social context of an increasingly urbanized and organized economy. Making plate armour required sophisticated workshop and guild organizations to divide labor, acquire and use capital and to maintain quality. This social context (which depends upon strong civil government and a codified system of contracts, among other things) is a necessarily precondition of technological developments like hammer-mills and polishing wheels - in many cases, these mills were funded by guilds or other craft organizations, or owned by proto-capitalist merchant-armourers like the Missaglia.
So plate armour was the product of all these things - larger blooms of steel, water-powered hammer and polishing mills, and more sophisticated social and economic organizations that could support the very sophisticated armouring industry. But we can't forget about the other side of the equation, the soldiers wearing the armour and their needs. As Tobias Capwell has argued, armour is adapted to the way that the people wearing it fight. English armours from the 15th century are meant for fighting on foot, while Italian armours of the same period are meant for fighting on horseback. This is reflected in the form of the armour. At the same time some features of armour - the neckline, the waistline, the shape of the sabbaton and more - are products of fashion.
When we see an armour in a museum what we're seeing is the product of all of this - technological developments, ways of war, economic organization and fashion.
If you are interested in learning more about the metallurgy and manufacture of armour, you may be interested in some of my previous answers: