France was humiliated by its loss of the Franco-Prussian War when Otto von Bismarck, the German Chancellor had imposed harsh terms, including the surrender of Alsace-Lorraine to Germany. Bismarck knew that he had created a volatile situation, and attempted to deal with it by a series of alliances designed to keep France diplomatically isolated. Among his other moves, the Berlin Conference of 1878 gave Austria the right to "occupy and administer" Bosnia and Herzegovina. No attention was paid to the nationalist interests of the people of the Balkans which were intense; rather Austria annexed Bosnia and Herzegovina into its own territory. When the Serbs became outraged, a series of skirmishes broke out. Austria easily put them down, and forced Serbia to cede Albania to Austria. In the meantime, the people of Serbia looked to Russia for support, as both were of slavic origin.
Germany was concerned at this time that if France attacked, it would be forced to fight both France and Russia, which was at its borders. This led to the development of the famous Schlieffen Plan which would allow Germany to fight a war on two fronts.
The assassination of Archduke Ferdinand was committed by a group of Serbian nationalists known as the Black Hand, who wished to remove Austrian influence in the Balkans. When Austria declared war on Serbia as a result, Russia mobilized, which it could not do without threatening Germany. This led to a declaration of war against Russia by Germany and subsequently against France.
The connection of the Franco-Prussian War to all of this is the series of alliances which Bismarck engineered in an attempt to secure Germany's gains from the war, and which ultimately pitted European powers against each other.