The question concerns itself specifically with its use on foot.
I know when it happened, a little bit of how it happened, and none on why it happened. If you look at early Polish swords from the Middle Ages, you can find a lot of your longswords and standard cruciform knightly (arming) swords. Then in the 15th/16th century the saber seemed to appear out of nowhere and managed to take Poland over.
I know that the development of firearms and the abandoning of armor had an effect, but the rest of Europe switched to broadswords (basket hilted straight sword), sideswords (between an arming sword and a rapier, also called a cut and thrust sword) and rapiers, both of which are (mostly) straight, double edged swords. The saber wouldn't start taking over the rest of Europe until the late 18th century and onwards when cavalry really started to develop. So how come Poland adopted the saber so early on, in favor of it, rather than straight, double edged swords, most of which have served Poland well for centuries?
As a sidenote, I don't believe that it's just "because of the cavalry". If that were the case then it wouldn't become the choice weapon for the Poles on foot. There has to be some kind of martial reason they switched, because I can't see any advantage of using a curved, single edge sword rather than a straight, double edged sword while on foot in a one on one.
Ok, so there are several things happening all at once here.
1st, and realize that this is something that is very important. "Because the Cavalry does it" IS a very common reason for foot to switch what they are doing.
One of the reasons for this is because the Cavalry are often drawn from the upper classes. And being a cavalry trooper was seen as a mark of status.
However, another reason is that the cavalry was often simply better equipped. Cavalry units often had more funding, and thus could afford better stuff. So, clever infantry men would... acquire such equipment when they could. This would often lead to leaders going ahead and adopting for the infantry later what the cavalry already had.
An excellent later example of this in the British army was the 1796 light cavalry saber. Which was so widely adopted by infantry that it led to the 1803 flank officer's saber.
Also realize that by the time sabers were being heavily adopted in Europe, first in Poland but spreading west from there, firearms are a thing, and heavy armor is dead or dying out.
Especially for foot. With that being the case, there is no reason to have a long, narrow blade for thrusting through armor for personal defense.
It is more use to have a spear or pike or firearm for your main weapon, and a lighter, more nimble weapon that is more suited to infighting for your secondary/self defense weapon.
The rest of Europe eventually switched to sabers as well. Because they are very effective for their purpose. Which is to be a good cutting sword suited to close fighting.
Realize that for military purposes, generally, a cutting sword is more useful than a thrusting sword unless you are going through or around armor.
The reason for this is simple. Most of the cuts are thrown to dictate movement and control space. This made the saber an ideal support weapon for other weapons such as the pike, the spear, and the rifle/musket.
The rest of Europe used other swords transitionally during this period due to those other weapons being a more direct natural evolution of the weapons they were using during earlier periods.
However, once the idea of sabers had been introduced, it grew steadily and by the 19th century pretty much every military in Europe and America, and even in some cases Japan, were using sabers.
Popularity of the saber among Polish nobility (and later Polish troops in general) is generally attributed mainly to the specific geopolitical location of Poland. The latter country bordered with Ruthenian and Lithuanian lands almost from the establishment of the country, retaining more or less friendly relations that later led to the union that created Polish-Lithuanian Commonwealth. To the south lied Hungary, another country Poland had good relations with. Ruthenians (also Russians), Lithuanians and Hungarians frequently used sabres (earliest specimens from Eastern Europe are dated to Russian Varangians from 9th century). As the contacts and cultural exchange between Poland and Hungary tightened at the end of medieval period, especially after Louis I, the king of Hungary, became King of Poland in 1370 and during a large influx of Hungarian, Transylvanian and Moldavian nobility granted lands by Wladislaw Jagiello who seeked new vassals and allies before and during the war with Teutonic Order in the early 15th century, Hungarian light cavalry (raci) became more and more common sight, heavily influencing clothing and armament of Poles. This also explains why the saber became popular among infantry, as Polish infantry units formed in 16th century were designed, to large extent, after the Hungarian infantry.
Furthermore, Poles had also frequent and way less friendly contacts with Mongols (Tartars) and later also Turks (predominantly Ottomans) what also led to adoptions of various elements of arms, clothing and decorations of Eastern provenience. Adoption of Eastern armaments and tactics was also necessitated by the requirements of warfare. Both Tartar and Turks relied heavily on light, fast and maneuverable cavalry, so Poles had to adopt similar formations to be able to intercept eastern enemies in the vast areas of south-eastern Poland. Use of tactics and weapons already proven to be successful by the enemies was a logical outcome.
Last but not least, ubiquitous use of sabre by the Polish nobility, forming a significant part of the Polish-Lithuanian military was cemented by the Sarmatism, a political and cultural national movement (although limited to the nobility class only) that focused on the mythical origin of the Polish nation tied to the ancient tribe of Sarmates and thus distinct Polish clothing and weaponry became not only practical items but also a political statement.