How did Niccolo Machiavelli connect the anarchy in international politics with his book The Prince?
Machiavelli approached his political treatise The Prince as a primer for rulers of various types of political entities extant in Italy during his time, the early 16th-century. He uses the term "states" to describe these political entities, which ranged from hereditary kingdoms to Republics and city-states found among the political amalgam of Italy at that time (which wouldn't become a unified nation until the late 19th century).
Machiavelli's overarching theme in The Prince is the notion that rulers, even those elected by democratic means (though those means were often restricted to election by a limited number of citizens at that time, mainly the powerful), could not rule in the abstract, i.e., they could not effectively rule via idealistic notions that involved broad participation in decisions by those who were ruled. Machiavelli inveighed against the notion that rulers should be transparent and entirely honest to those who were ruled, and in fact often must adopt subterfuge and blunt political stratagems that might go against the popular will, for the benefit of not only the state he ruled, but for the subjects of citizens of that state. A famous quote from the work is the following:
"He who neglects what is done for what ought to be done, sooner effects his ruin than his preservation."
Machiavelli observed a world that, as in many parts of Europe such as Germany, possessed myriad political, religious, class and racial tensions, tensions that often threatened to boil over and result in great upheaval. His contention in The Prince was that political rule required a strong hand to control these tensions and keep competing factions in check, and that the means of controlling political turmoil was justified by the ends, i.e., a modicum of political tranquility.