There were at least two ways in which the international climate contributed to German unification.
First, the international community was becoming more used to the idea of nationalism. The push for nationalism and nation-states had been building for some time. It had helped to create a unified Italy just before Bismarck came to power in Prussia. Part of the international climate, then, was nationalism and this helped the cause of German unification.
Second, German unification was achieved largely through wars that were made possible by the international climate along with skillful manipulation on Bismarck’s part. The first war that was relevant was Prussia and Austria’s war with Denmark over Schleswig-Holstein. The Danish incorporation of those duchies was a part of the international climate that allowed a German war to take Schleswig-Holstein away from Danish administration. The second war was between Austria and Prussia. Bismarck maneuvered Austria into this war while simultaneously getting Russia and France to remain neutral. The Austrian willingness to fight and the Russian and French willingness to stay neutral were all part of the international climate that allowed this war to occur. Finally, a war with France that was ignited by a dispute over the throne of Spain finished the unification. The presence of monarchy and the willingness of countries like France to try to help determine who took the throne in other countries helped allow this war to come about. In short, Bismarck was able to take the attitudes of various countries and manipulate them to bring about wars that allowed the unification of Germany.