What enabled Germany and Italy to create unified countries in the 1800s? How did this force later contribute to the start of WWI?

During the early nineteenth century, nationalism and ethnocentrism became central to the thinking of European intellectuals and, ultimately, the peoples of Europe overall. This was especially true in the German and Italian states, which had been repeatedly invaded and subjugated by Napoleon's forces. The Germans, especially, felt humiliated by their having been reduced by the French into client states. Germans and Italians alike already had a mythic sense of their past greatness and also a sense of failure in not having already created unified nation states as the English, the French, and the Spanish had done long before.
As a result of the final settlement at the Congress of Vienna in 1815, Prussia became a major power, partly due to its having shared with Britain the achievement of defeating Bonaparte's final stand at Waterloo. This put Prussia in a position to wield power and influence over the other German states. Prior to the Napoleonic wars, if there was any claim to leadership of the German peoples, it was that of the Hapsburg monarchy in Vienna, since this was the seat of the largely theoretical Holy Roman Empire. After 1815, Prussia was more than an equal rival to Austria, and Prussia in fact became stronger as a leader of the Germans. Because Austria ruled a multi-ethnic territory, it was not in a position to adopt a central role in the creation of a specifically German nation state. The Austrian leadership also had its own internal problems with the Hungarians, who succeeded in forcing them to create the "dual monarchy" of Austria-Hungary, placing the Hungarian leadership on a presumably equal footing with the German-speaking ruling house and nobility.
The Prussians took advantage of these factors as a whole, and the key event was the Austro-Prussian War of 1866. Bismarck, the Prussian chancellor, succeeded in gaining alliances among the German states and paving the way for a unified nation state that would leave out Austria. His defeat of France in the Franco-Prussian War also allowed the Germans to gain previously disputed territory. The finalization of a unified Germany in 1871 was what the Germans referred to as Kleindeutschland (small Germany), as opposed to a Grossdeutschland (greater Germany) that would have included Austria. Presumably the latter would have been possible only if the Austrians had given up their non-German-speaking territories, which they were not prepared to do.
The situation in the Italian states was more complex because of the greater factor of foreign domination. In 1815 Austria held parts of northern Italy, including Venetia and Lombardy. Given the decreasing strength of Austria and the nationalistic fervor taking place in the intellectual world, it was inevitable that this would change. For the Austrians to be expelled and for the Italian principalities to be brought together, leadership was needed on several levels. King Victor Emanuel II of Sardinia-Piedmont became the face of a unified Italy, so to speak. His minister Count Cavour was to some extent the mastermind, and Giuseppe Garibaldi was the military genius required. By 1861 the Italians had succeeded in bringing most of the peninsula under Victor Emanuel's leadership. In some sense, the last obstacle was the resistance of the Papal States. The entire unification process, finalized in 1870, in which Rome also became part of the new Italy, would probably not have been possible without the increasing secularism of European thought in the nineteenth century. Garibaldi was an agnostic, as was much of the political and intellectual elite of Italy and Europe as a whole. What had been thought impossible—depriving the Pope of his temporal power—became a reality.
The unification movements, especially that of Germany, had grave consequences for the stability of Europe. Now, France and Britain began to see the new German state as a danger, a rival to their own power and their control of colonial empires. France wanted revenge for its defeat in the Franco-Prussian War, and both France and Britain wanted to ensure that Germany would not take over additional European territory or territory in the Middle East, the weakening Ottoman Empire, and thus gain control over the newly built Suez canal and the trade routes to the East. Europe became a tinderbox, and the explosion came in 1914 with the Great War.