Cannons certainly played a role, but they weren't the whole story.

The transition from feudalism (itself a nebulous concept) to early modern nation-states is a complex one which can be attributed to a variety of causes. It's probably best to deal just with the most relevant feature of the transition being discussed: the development of the centralized state and the elimination of competing regional power-holders.

 

Feudalism, to paint an infinitely diverse system with a broad brush, was a method of utilizing the revenue of a country's lands without the need for a complex central bureaucracy. Rather than having an organization of royal tax collectors to assess and collect revenue from all the villages in the kingdom, the land would be granted to someone in exchange for their fealty and military service in times of war, in Europe often on a hereditary basis. That person would then handle revenue collection at the local level. Europe, during much of the Middle Ages, had a relatively cash-starved economy. This meant that villagers could typically not pay all of their taxes in cash, and paid by giving away a portion of their produce to their lord ("paying in kind"). For obvious reasons, this would have made it very difficult to maintain a central tax-collecting bureaucracy for the entire country, since financial clerks in Paris would have found it nigh impossible to (for instance) turn wheat collected in the south of France into anything of use hundreds of miles away in Paris. Hence granting the land to local lords who, being local, could actually use the wheat or more easily market it to turn it into useful cash. Over time, this became entrenched as a complex network of overlapping rights and obligations.

So feudalism was to some degree a method of dealing with the relatively cash-starved nature of the European economy at the time. Once cash became more plentiful, states again had the capacity to centralize, as revenue collection on a large scale became possible. It was the infusion of cash into the European economy, not cannons, which made the end of feudalism a possibility - indeed those cannons also would not have been able to be deployed in large numbers had Europeans not had the cash to pay for them.

 

However - cannons did play a major role in state centralization, once the process began. All those local lords had to be suppressed somehow, and cannons allowed the expanding state to more easily crush their resistance and destroy their innumerable fortified castles. The guy making the video seems to miss that crucial point: cannons made centralization easier by enabling kings to defeat potential internal resistance.

 

I'm also skeptical of the idea that the appearance of cannons directly led to larger army sizes. As far as I'm aware, the largest relative increase in the size of European armies came during the sixteenth century, after the type of centralization referred to in the video was already well developed in the major states of Western Europe - also a period when fortification technology was rapidly adapting to be able to resist cannon barrages, the so-called trace italienne (see for instance Black, Jeremy. European Warfare, 1494-1660. Routledge, 2002). This YouTuber seems to be addressing fourteenth and especially fifteenth century developments, before the rapid expansion of the size of European armies associated with the Italian Wars.