What does "new royal power centers" mean?

User Evargalo's answer is essentially correct, but I'd like to elaborate a bit.

First, note that the base sentence has somewhat unusual syntax because of the inversion: it's "draw to new royal power centers the resources of agriculture and trade." The thing being drawn is the resources; they are being drawn to "royal power centers." Hopefully that makes it a little easier to parse. The "royal power center" then just means "the capital," since that's where the king has the most direct power/control.

What Spence is describing is the process of state formation and centralization. Prior to the period in question, governments through much of the world were highly fragmented: a king might nominally rule over a whole country, but his royal power would be very limited outside of a central holding that he could rule and administer directly. So if you look at maps of the Crown Holdings of the King of France, for most of its history the king only directly controls maybe 20% of the country. Many of his dukes actually have far more extensive holdings--so to rule, he has to rely on their cooperation and goodwill. He has to keep them appeased, or they'll revolt, and in many ways he's at their mercy. After all, they are the ones who control the "resources of agriculture and trade"--not to mention the armies the king might use to enforce his will. In any event, particularly in the European context, the king didn't really have the bureaucratic resources to enforce his will elsewhere anyway: stuff like issuing laws (drafting them and telling people what they are), administering laws (i.e. running a legal system/courts), collecting taxes, etc. requires a lot of highly educated people to act as scribes, accountants, negotiators, etc. With the decline of the Roman Empire in Europe, those people were in very short supply--that was probably the biggest motivator for feudalism, and it's why you see systems such as tax farming, labor taxes, or taxes on specific commodities--in a European or Chinese context--it's just much simpler to administer those things given limited state capacity.

Not to mention other challenges of running things from a central city in the days when the fastest communication method is "send a guy on a horse"--the king in Paris (or Emperor in Luoyang) cannot respond dynamically to events far away when he may not even hear about them for several weeks.

That said, centralized power is obviously desirable for the people who would hold the power, and centralized states can accomplish a lot more than fragmented ones. So one of the big stories you can tell about the period from say 1500-1700 is of the king/czar/shogun/etc trying to get more direct control, remove second-level competitors for power (big landholders, dukes, etc), and develop the kind of systems that will let them make decisions from the capital and see those enforced.

It's been a while since I've read The Search for Modern China so I don't remember the exact argument here, but this is actually a problem that China had been grappling with--often quite successfully--for a very long time. The whole point of the Imperial Examination System is to create a cadre or class of highly educated bureaucrats who owe their allegiance to the emperor (or at least the imperial bureaucracy) directly: people who have the skills to do administrative work, and don't have a local power base with which to fight the central administration. That's also why they were moved around so much: see rules about reassignment of magistrates so they wouldn't work in their hometown and wouldn't hold a particular regional post for more than a few years.

In other places--like Europe (Louis 14 of France, several generations of Habsburgs starting with maybe Frederick 1 of Austria, Joan 5 of Portugal, and others), Russia (e.g. Peter the Great), or Japan (Tokugawa Ieyasu)--leaders were in this period finally reaching a point where they could establish stable central governments, either through war, commercial dominance, or sheer force of personality (to get junior nobles to go along with it). That process typically involves setting up or expanding a capital and making sure that everybody in the country knows the capital is Where It's At, not just politically but culturally and economically and socially. That's your "royal center of power," and the process Spence is talking about is the process of figuring out how to effectively move material wealth around so that this can happen.

Just my interpretation - more context would be useful and may prove me wrong:

Around that time many regions that lived with a feodal-ish style of organisation came under the control of more and more centralized powers that will become real nation-states in the next century.

For instance, Habsburg monarchy in Austria came to rule over provinces in Hungary, Silesia or Northern Italy that were used to some autonomy in their economic life. Bringing the resources from such areas to Wien, by the mean of taxes, royal administration or trade rules, has been a long and non-obvious process over many decades.

The same must apply to Tokugawa's Shogunate in Japan, the Mughal Empire in India, and to a wider extend to all the colonial Empires, Spain and Portugal notably.