Recruitment Centers are buildings that increase the target city's manpower gain by 25%. These are best used in large cities like Tokyo or New York. Unlike developing cities which does the same with the city's manpower gain, a recruitment center can only be built once in a city.
Recruitment Centers cost differently to produce according to the population of the city that you are building them in.
Building recruitment centers does not increase your manpower capacity, it increases your manpower gain. You can find the exact amount of a nations manpower gain by hovering over the bottom part of the screen where it says "Manpower".
It is a wise idea to destroy recruitment centers in your cities if they are being taken over by an enemy and there is little-to-no possibility that the city can be recovered, as recruitment centers can also benefit your enemy's manpower gain.