This is certainly a complicated question, but it’s also a very interesting one that arguably undergirds the history of modern Europe in general. Before answering though, it's important to acknowledge that the question itself has a long, fraught history. Versions of this question were already being asked and answered by European thinkers as early as the 1700s and 1800s. These thinkers, who include figures like Adam Smith, Max Weber, and Thomas Macaulay made some useful observations about early capitalism and the nature of European states. However, since they were writing from within empires, they often ended up trying to justify imperialism by arguing Europe had become more powerful because it was more “civilized” in terms of religion, race, and governance. These are not factors that reputable historians would cite today, but they have infiltrated historians’ understanding of the question in ways that are not always immediately obvious. Today, historians try to be on guard against this type of “eurocentric” thinking, or the idea that Europe is inherently superior to the rest of the world. One good why to do this is by being specific about the question and its parameters. So, my answer assumes that by “powerful” we mean nations have large economies, strong militaries, and therefore are able to “control” much of the world by force. I also am going to focus primarily on western Europe (Britain, Spain, France, Netherlands, and Germany) since those are the nations that do develop large overseas empires.

The first question is when did European nations become more economically and militarily powerful than the rest of the world. There are three eras we might consider here. The first is the 1500s- 1600s, when Spain, Portugal, Netherlands and Britain begin to expand overseas: Spain finishes the Reconquista, which expels the Islamic Nasrid dynasty from the Iberian peninsula (1492); Cortes conquers Mexico City/Tenochtitlan (1519); a Spanish-Italian fleet defeats Ottomans at Lepanto (1571); the Dutch-India company is founded (1602); and the British East Indian Company secures forts through the 1600s. Until this point, Europe was largely peripheral to a wealthier, more interconnected, and more scientifically and technologically advanced Asian world. (For more on this, see Abu-Lughod, Before European Hegemony) Through the 1500 and 1600s they seem to be developing into strong competitors for existing empires. However, there’s debate about to what extent Europeans were actually more powerful in this era. One point raised by south Asian historians is that a lot of European historians rely mostly on European primary sources. They argue European narrators are especially unreliable in this era because either (a) they didn’t speak languages or understand culture enough to get what was happening at all or (b) they exaggerated their successes to get more funding from patrons. Other historians have argued that European gains in this period came from being in the right place at the right time rather than being more technologically advanced: that powerful Asian empires like the Ottomans and Mughals were suffering from internal issues that made them temporarily weaker. In the context of the Americas, a similar argument can be made about disease temporarily weakening indigenous civilizations. (sources: Subramanyan, Courtly Encounters, 2012; Casale, "Global Politics in the 1580s," Journal of Global History, 2007)

The second potential era for Europe gaining power is the 1700 to the mid-1850s, when European countries gradually took more land in Asia and the Americas. However, it is important to emphasize that this growth is gradual: European expansion is successfully stopped or slowed by indigenous groups in all these places. For example, it takes British forces almost a century to fully conquer the Indian subcontinent, and in North America the U.S. is fighting wars across the Great Plains through the 1800s. As late as 1881, British forces were defeated in Egypt by a Sudanese force at the Siege of Khartoum. So through this period we could say European power is growing, but is still not absolute. (some sources: Bayly, Imperial Meridian, 1989; Jasanoff, Edge of Empire, 2005; West, The Contested Plains, 1998)

The moment Europe’s power really becomes clear is the so-called “scramble for Africa,” the period from 1885-1910 when the entire continent of Africa is rapidly colonized by European countries after the Congress of Berlin sets terms between European countries about what counts as a “legitimate” colonization. (Obviously African nations are not part of the Congress of Berlin, which makes it an odd sort of agreement.) At this point, Western European countries and their former colonies in the Americas seem undeniably powerful because they are able to take over so much land so quickly. (source: Burbank & Cooper, Empires in World History, 2010)

Based on this timeline, most historians argue European power must be tied to industrialization, since eras of intensive colonization basically correlate to the first and second industrial revolutions. Good evidence for this theory is that Britain, the country that industrializes first, also ends up with the largest empire. Industrialization creates larger economies, more advanced weapons, and enormous population growth, which allows European empires to conquer territory and rapidly settle it. Europeans’ success in rapid conquest during the “scramble for Africa” seems largely due to technology (steamships, guns) and medical advances. So the question then becomes why did Europe industrialize first? (source: Headrick, Power Over Peoples, 2010 and Satia, Empire of Guns, 2018)

This is where different historians have different answers. One answer has to do with location and resources: that Europe (especially Britain) had access to easily mined coal reserves, which allowed them to start producing more, which supported a large population, which allowed them to produce more, etc. Historians have also noted Europe has a more stable climate and fewer natural disasters which lets them produce more food, leading to more people, producing more food, etc. In the 1700 and 1800s, population size is a huge part of economic potential, so expanding agriculture is really significant. (main source: Pomeranz, The Great Divergence, 2000)

Another answer has to do with scientific culture: that something about Europe makes scientists slightly more innovative than elsewhere. One argument here is that Europe’s small, warring states let unorthodox thinkers escape persecution, leading to more innovation. Another is that Europeans were more focused on “useful” invention, they took abstract ideas (possibly from elsewhere) and figured out how to apply them to "practical" machinery and weaponry. (main source: Mokyr, The Enlightened Economy, 2009)

Another answer that’s a bit more specific to Britain is the idea that consumer demand pushed industrialization (the “industrious revolution.”) (source: de Vries, The Industrious Revolution, 2008) Others have also argued the British state in particular developed strong “protoindustrial” infrastructure. In these cases, Britain led the breakthrough and then exported industrialization first to Europe and then elsewhere. These explanations don't rely on a single factor, instead they describe a series of gradual shifts that puts Britain in a more stable economic position than the rest of Europe.

From that point out, its basically a head-start argument. Nations that were never formally colonized, like China and Japan, do industrialize and form their own empires in turn but are just a step behind Europe (until, arguably, now...) North America and white settler societies (Australia, New Zealand, South Africa) do catch up eventually, the U.S. surpasses Britain after WWII to take global superpower status. Meanwhile, colonized countries suffer more significant setbacks because European empires essentially de-industrialized them when they occupied them with lasting economic consequences.

Hopefully that’s a helpful start on a question that is the subject of entire courses and careers. You can see how this rapidly becomes very complicated: there are a lot of factors to consider, a lot of different regions and nations you can focus on, and a fraught history of answering it. Happy to elaborate on any of the above.

(edited to add some more sources)