The European powers were able to conquer such broad territories so quickly because of their technological superiority. No country that got colonized had anywhere near to the technology (or the economic power that comes with technology and leads in a spiral to more technology) that the Europeans (and Americans had).
As an example of the power of technology, think about how China and Japan reacted to the West. China tried to turn inward and keep the West out. Japan, instead, had the Meiji Restoration and industrialized. This is the major reason why Japan became a power and China was essentially colonized.
The British Empire far exceeded any else during this time frame. All other countries were attempting to replicate or compete with what England had done. The Continental Powers, notably Germany, had recently unified after centuries of "Balkanization" and the resulting Nationalism created, in Germany, at least, a desire for a Germanic Empire, stretching from Germany proper through the Middle East, in addition to colonies in Africa. Much of Europe cast influence to Africa, simply because they could as organized industrial powers, and if they did not, would be helping other European colonizing efforts by default. So the race was on. The increasing conflicts along colonial borders eventually came home in 1914, and Europe tore itself apart.
There was also intense competition between empires at this time. No less than the Germans, Russians, British, French, Austro-Hungarians and Ottomans were trying to expand their territory and influence. For most of them, this meant Africa and Asia would be the colonial targets.
This competition made them move more aggressively to colonize than they perhaps would have normally, and gave them a short term advantage when combined with the technology. There was a rush before all the profitable, attractive colonies were gone, in other words.
These were also nationalist empires, and quite racist. While not an advantage directly, just as with competition, it was a powerful motivator and justification for their actions.
The convergence of technology and unabashed industry helped to enhance imperialistic claims at the end of the 19th and start of the 20th centuries. Advances in technology allowed greater machinery to help deliver imperialistic claims and hopes into reality and also assisted in subjugating those who opposed it. The proliferation of capital and industry helped to not only finance imperialism, but start the process of setting up satellite branches of companies in other nations. Additionally, the income growth helped to influence government policy and practice with dollar diplomacy and ensuring that imperialistic practices were in line with proper business techniques. The collusion of both big business and government action ensured that imperialistic practices could be carried on a larger and more enduring scale.