I have basic understanding of the Franco-Prussian War. It started because Prussia and France were arguing over the selection of Prince Leopold as King of Spain. Bismarck altered a telegram to paint the French as insulting King Wilhelm and so, France declared war on Prussia, which rallied the other German state behind it to create the German Empire.
But, I want to know why Germany, after having decisively crushed France, decided to annex a small slither of land like Alsace-Lorraine. What was the reason behind this annexation?
There were multiple motivations for Bismarck to annex Alsace-Lorraine to the new German Empire, but the major ones that I can see are the following:
Nationalist and historical reasons - Friedrich Engels, in a note dated August 10 1866, wrote that there was a "national fury" engulfing the North German states to annex Alsace and Lorraine, which contained German-speaking populations. If Bismarck were not to heed this anger, he would be greatly unpopular, though at this time of writing, historians like Fritz Stern, in his book Gold and Steel argue if Bismarck was a victim or the cause of this "national fury". Also, the duchies of Alsace and Lorraine were considered traditionally German, since they were instituted under the Holy Roman Empire, though the two duchies would come to be annexed and partly bought by France, first in 1469 in the Treaty of Saint-Omer (Upper Alsace was sold to the Duke of Burgundy whose nominal overlord was the king of France), then around the 1630s - 1640s (Habsburg Sundgau), and then by 1648, both duchies were finally confirmed to be France's in the Treaty of Westphalia.
Resources - According to Hartshorne, one easily-justifiable reason as to why the Alsace-Lorraine region was annexed was due to Lorraine's considerable iron deposits. However, he also notes that this a small concern, compared to the next one.
Military - This was the biggest consideration that affected not only Bismarck, but also then-Prussian army marshal Helmuth von Moltke the Elder. Considering that France to be their mortal arch-enemy, most of the direct borders the new German empire had with France would be the area of Alsace-Lorraine, which contained vast fortifications that would make any assault into South Germany easy, such as the fortresses in Strasbourg (which controls the easy passes in the Vosges Mountains) and Belfort (which controls the gap between the Vosges and the Swiss mountains). By gaining control of these areas, not only does the Germans deny these easy means of invading Germany but also allow them to defend against attacks by France more easily. This is also the reason why the primarily French-speaking area of Metz was also annexed, since it contained the Metz Fortress, the one good fortress stopping any advance at any other point along the Moselle river apart from the Vosges.
Unification - Due to the reason above, Bismarck also needed Alsace-Lorraine to make sure that the southern German states of Baden, Wurtemberg and Bavaria would join the new German Empire. According to Ludwig Hahn, Bismarck learned from the then-King of Wurtemberg, Charles I of Wurtemberg, that it was the French control of Strasbourg fortress that essentially is stopping the southern German states from siding with Prussia on the matter of German Unification. With not only Strasbourg fortress controlled but all major passes along the Moselle, the southern German states of Wurtemberg, Bavaria and Baden were now confident in joining the German Empire without fearing reprisals from the French.
Sources:
*Stern, Fritz. Gold and Iron. p.139 *Hartshorne, Richard. “The Franco-German Boundary of 1871.” World Politics, vol. 2, no. 2, 1950, pp. 209–250. JSTOR, www.jstor.org/stable/2009189. *Hahn, Ludwig. Furst Bismarck: Sein politisches Leben und Wirken