The phrasing of this question makes it enormously complicated. I'll talk about three levels, and hopefully other redditors can jump in as well: (1) How widely did the 16C recognize the idea of a "better" world (2) Differences in 15C versus 16C life for "commoners" (3) What kinds of information would 16C "commoners" have had about the recent past. I put the word in quotes because I'm not sure quite what meaning that holds. This answer will use Germany/the Holy Roman Empire (HRE) as a proxy for "central Europe" and discuss primarily the rural peasantry.
Medieval Latin Europeans dwelled in a "world grown old," a world decaying towards the arrival of Antichrist and Armageddon. When 12C "renaissance" scholar John of Salisbury cited Bernard of Chartres as saying "If I have seen so far, it is because I have stood on the shoulders of giants," the point was that the age of the giants was over. While this is the elite of the elite, the apocalyptic climate in the 15th century ramping up to the Reformation was even more intense. A point that often gets lost in modern conception of "reform" as improvement for the future is that the Reformers acted out of a zealous belief that they faced an imminent Judgment Day. They worked to save souls in a world that would only get worse.
But a shift occurs over the 16C. In postils--collections of sermons--the interpretation of natural disasters like plague and flood shifts away from APOCALYPSE SOON! Preachers continue to stress the need for repentance and prayer (now along proper Lutheran/Calvinist/Catholic lines), but they stress that prophecies of, for example, disastrous popes have been fulfilled, not loom as Antichrist in the near future. 16C sermons offer comfort and consolation in a more heightened, systematic, and, I say as a medievalist, successful manner than 15C ones.
Postils, as David Frymire points out, are a key source for triangulating (if not recording) "popular" views. These books were widely printed and owned by even rural parish priests, and disparaging remarks by the Postille authors tell us that the sermons were preached verbatim at least sometimes (rather than being guides). The goal of providing consolation and appealing to the audience was baked into the sermons, and indeed, sermons were massively popular events in the 16C. People clearly responded on an emotional level to the messages they heard, which suggests that these texts offer at least a frosted window into a broader 16C mentality than just their writers'.
You could even, I think, point to the demands of the 1524-25 Peasants' War (PW/GPW) armies as a recognition that the world could improve (or at least, could return to earlier status). Again and again, we see the call to return to "God's law" as the rightful determiner of water use rights, hunting rights, fishing rights. Peasants protested what they saw as unjust, recent developments. It's common to attribute the more "optimistic" mindset to the so-called Renaissance, whose scholars viewed themselves as illuminating the 'dark ages' with the light of classical antiquity. But this was a broader development on multiple levels of society, that happened to include the vaunted Renaissance.
So if 16th century people had the basic concept that the world could improve, the questions follow: what was different, were the differences "improvement" (cui bono?), and who would have been aware that these things were different.
The PW, indeed warfare in general, is a good introduction to the difficulties of talking about "improvements" in 15-16C HRE. The PW was massively destructive, to be sure. But was it the last of the great medieval anti-feudal revolts...or the first of the great anti-territorial-state-apparatus revolts? There is a certain tinge of "meet the new boss" in the historiography on the matter.
On wars more broadly, the 16C certainly produced more name-recognition conflicts. The rise of confessional (religious) warfare did not spare the HRE, although the Peace of Augsburg helped either delay the worst until the next century or made the worst even worse when it finally happened. But the 15-early 16C witnessed the bloody, brutal process by which the errant "robber knights" and petty nobles were brought under the dominion of the territorial princes, the unofficial, constant warfare in which peasants were as always the victims.
For the rural peasantry in western Germany, there were perhaps fewer developments than one might expect given the enormous cultural and martial upheavals of the early 16C. There was a general shift away from viticulture across the HRE, with cereal agriculture taking its place. Tom Scott observes that areas suitable for growing grapes continued to grow grapes, and suggests that this change reflects (1) the need to feed a growing population (2) better recognition of crop/soil/climate fit. One result of this change was that Germans drank less wine and more beer. So, good news for early modern Oktoberfest?
One innovation was the formation of guild-type organizations for rural craftsmen in the orbit of German cities. But this is an ambivalent development. On one hand, it provided these artisans with a collective voice in the local economy, perhaps better access to urban markets. On the other, these were a reaction to stronger urban craft guilds that tried to limit rural artisanal competition. So definitely a give and take situation. As cities expanded in both size and fiscal power, city-country cooperation and competition grew increasingly important and increasingly contested. Apparently the situation sometimes grew dire enough/threatened the overall economic situation enough that territorial princes got involved where they could. (Not all HRE cities were subject to princely authority.)
Another HRE-wide development was the increase in mining and mining-based towns. This is still going to represent a small fraction of society, and I'm not sure you can call it an improvement in terms of a miner peasant versus their earlier or contemporary farmer counterpart. On the other hand, as a result people would have had...saltier food? More metal implements? (Iron and tin-plating perk up tremendously in this era, especially around Nuremberg in Bavaria). The spread of the glass-making proto-industry meant better-off peasants might be able to acquire glass for their windows, too.
But while the general fabric of rural society remained fairly constant in the western HRE, eastern peasants weren't quite so lucky. Eastern population growth and urbanization lagged behind the west. Eastern HRE economies oriented towards commercial agriculture serving export markets--in the western HRE, other European destinations, and even overseas. The drivers and beneficiaries of this organization-reorientation were not peasants or burgers but the landed nobility. There was a gradual erosion of earlier peasant status as producers with the right to hold land and determine its use. Lords exercised the control. This shift was not a new creation of the 16C, to be clear, but it was accelerating in the east even as peasant status crept upwards or held its ground in the west.
But would peasants have known about these changes? If so, which ones? We saw some signs in the Peasants' War that people had the idea things were different in the fairly recent past, although one could posit this was simply a rhetorical stance to achieve goals like better resource access. So how might rural peasants have learned about their counterparts in the recent past?
Outside the cities, literacy rates remained rock-bottom low. However, oral tradition, preaching, and the occasional reader would have been sources for recent historical knowledge. (Separate from ideas about biblical times or ancient Greece). When William Chester Jordan looked at the future legacies of various 14C disasters, he found that the Black Death stuck in people's minds in a way the Great Famine did not. We know that plague and famine reoccurred in waves after that; plague being quite fearsome but never as disastrous. In fact, there is some evidence to suggest that either early modern people got better at combating plague through quarantine, or perhaps the virus evolved to be less catastrophically fast-killing. So would the 1350 Black Death stick out in 16th century minds as "that horrible time in the past"?
In fact, what we see in 16th century postils is the opposite! A universal view that the world after 1550 or so is a degenerating mess of plague, famine, war, catastrophe after catastrophe, permeates the sermons. This is not pastorly melancholy. 1560-1630 was another 'little ice age' across central Europe, bringing with it shorter growing seasons and weather-induced bad harvests. In combination with a population on the rebound after 1470 (and not entirely devastated by the 1524-25 Peasants' War, to which this answer will return in a bit), the result was less food for more people. If people went to bed hungry each night, it must have been tough to feel that they were living in a "better world" than their 1450 ancestors.
Further reading:
Related question: Is the mindset that we live in a "better" time a modern one? So would someone in the 1500s even view any sort of technological progress as being good? As in the mindset that the non-existent past was some golden age.
Perhaps the question could also be answered the other way around, "why would someone in the 1500s think someone in the 1400s was better off?".
When you say advancement, are you speaking strictly technological or other (religious, political environment, etc?)
Because what we would be looking for technology wise, wouldn't be when something was invented. Instead we would look for when a technological advancement actually "trickled down" to European commoners.
Example: The "modern" flushing toilet was invented in 1596, but didn't become really widespread until maybe the late 19th century.
Source for the modern "water-closet"creation, Kinghorn, Jonathan "A Privvie in Perfection: Sir John Harrington's Water Closet".
Edit: I did not add this in just because of the immense complexity in the answer (as in entire books are written on the subject) but now it seems it should at least be mentioned.
The technological advancement that would most likely hit commoners first would be warfare technology. Following the progression of not just cannons and gunpowder from Asia, but also the ability to forge better combinations of steel and iron, and how the build of different weapons changed through these years is something that a number of historians look at today. However despite this most likely being the technology with the greatest geological and economic diaspora, it was not (as others have discussed on this thread already) something that commoners would necessarily be upbeat about.
I would reference the reactions soldiers had to the new weaponry (particularly chemical weaponry) in WWI/The Great War as the standard for how commoners would have reacted to the appearance of canons; these would NOT be happy reactions. In fact it would most likely be reactions of horror to what would befall their fellow commoner soldiers and themselves.
Sources: Reddy, Chris "The Growing Menace of Chemical War"
Maurice Keen "Medieval Warfare: A History"
Edit 2: Grammar and spelling.
Although a late 15th century invention the flyer on the spinning wheel was very impactful on everyday life. It looks like the flyer came into being around 1470's and was everywhere by the early 16th century. For those that don't know what a flyer on a spinning wheel is, it's this thing. Prior to that, anyone using a spinning wheel had to hand walk the thread around the bobbin and start spinning again. Talk about a sudden increase in production!!! No more stopping, winding, and then spinning. You could sit there and hand crank out all the thread or yarn you needed until you ran out of wool/flax/cotton/ect. Here is a late 15th century depiction of a spinning wheel with the flyer at the far end. The treadle wheel - where you use your foot to turn the wheel - won't be invented until the 17th century.
