The question is somewhat malformed, because there is no actual proof that Bismarck ever said that. In fact, I was surprised to read all about it a few years ago when I went from the German-language historical sphere to also discussing history on an international scale in the English language - because we Germans have no record of this quote, and one of us is supposed to have said it. Seeing as how English was not the international language of choice back then, it would stand to be argued that there should be more German sources about this quote than English ones - but in fact, there is not a single one that I know of. If anyone knows the German original, I'd be glad.
But as they say, absence of evidence is not evidence of absence, and there certainly was a whole bunch of conflict in the Balkans and concern about said conflict in the rest of Europe. So this is the supposed quote, in English, as according to Wikiquote's "Disputed" section on Bismarck.
Europe today is a powder keg and the leaders are like men smoking in an arsenal … A single spark will set off an explosion that will consume us all … I cannot tell you when that explosion will occur, but I can tell you where … Some damned foolish thing in the Balkans will set it off.
Wikiquote takes this quote from Andrei Navrozov's Chronicles (Volume 32, 2008), with Navrozov also already putting its legitimacy in doubt. The earliest apparance of something akin to this Bismarck quote in English language literature seems to be Winston Churchill's 1923 account of the Great War, World Crisis, more specifically Volume 1, concerned with the years 1911 to 1914. Churchill attributes the quotes as having come to his attention from Albert Ballin, a German diplomat, who in turn according to Churchill told Churchill that he heard it from Bismarck. Dubious, to say the least.
So the first time someone mentions the quote is 1923, a quarter of a century after Bismarck's death - and that someone is Winston Churchill, a man who always was more concerned with writing history with his books than writing books about history.
Anyway, with the question about the accuracy of the quote out of the way, we can nevertheless look at the history of the Balkan Conflict up until the 1890s and then until 1914 and beyond.
First, a lesson in historical geography. This map comes courtesy of the great cartography index of the University of Texas at Austin. Seriously, UT has amazing maps from various historical eras freely available to browse. Bless them.
Anyway. The map is from 1899, so shortly after Bismarck's death and the last possible nsituation he could have looked at. As the quote is IMO bogus, we might as well go to the 1890s rather to the 1870s.
The Balkans are surrounded by four great powers:
The Ottoman Empire, to the southeast and in the center. The Ottomans for a long time were the principal power in the Balkans and defined most of its conflicts. It was the Ottomans who brought Islam to the region, which is the reasion Bosnia and Herzegovina and Albania are majority- or plurality-Muslim countries to this day. This religious zeal on the part of the Turks made them easy to hate for the local Christians, be they catholic, orthodox or protestant. However, the Ottomans were for a long time the absolutely dominant power of the region, easily suppressing any and all resistance.
After taking Constantinople in 1453, they swept across the disunited and squabbling Christian warlords straight into Hungary, seizing parts of it for themselves and only being stopped when their first siege of Vienna was repelled in 1529, the first of several painful defeats for Ottoman expansionism, next to the loss at Lepanto in 1571, the failed attack on Malta in 1565 and eventually their second failed attempt to take Vienna in 1683 [insert Sabaton meme here]. But although their expansion was stopped, the Christian reconquest of these territories was slow and riddled with losses at Turkish hands. The primary antagonist to the Ottomans were the Habsburg Austrians, to whom we will get shortly. Through the 1600s and 1700s, the Ottomans remained a force to be reckoned with, but through various, primarily internal, factors, they began to decline in the late 1700s.
After initial Ottoman victories against Serbian insurgents, the country successfully broke from the Ottoman yoke in 1817. In 1821, the Ottomans failed to prevent Greek independence. The 1850s saw Montenegro break free and when Russia and the Ottomans found themselves at war in the late 1870s, that also provided the chance for Romanian freedom. The Ottomans by 1899 were on their way out - but they still held the heartland of the Balkans, including various peoples that still desired freedom or that were appealing to control for the other great powers.
These other great powers being the Austrians to the northwest and the Russians to the northeast and to a lesser extent Italy across the sea in the west. Italian ambition primarily aimed at North Africa for now, going to war against the Ottomans in 1911 and critically seizing not only Libya but also several islands in the Aegean Sea, but Italy would by World War 2 also get involved in the Balkan game. That's a story for another time.
So, what did these powers have in terms of interests and conflicts at the time of Bismarck's death?
The Ottomans wanted to survive. Cutting their losses and preserving what they at that point still had was their main objective, with a counterattack not really possible. The Balkan Wars of 1912 and 1913 would of course prove them right, being pushed off the Balkan peninsula entirely until Bulgarian arrogance provided them a chance to at least restore control over the area of East Thrace - a territory that to us is better known as "that weird part of Turkey in Europe even though most of the country is in Asia".
The Austro-Hungarians wanted to expand. Not only did this rampant Austro-Hungarian anti-nationalistic imperalism eventually cause the first declaration of war of World War 1, Austria-Hungary saw itself threatened by its various ethnicities (and rightfully so) as well as by the surrounding great powers. Both Italy and Russia were perceived as threats (and rightfully so). It was opportunism after losing the struggle regarding German dominance to Prussia in the 1860s that primarily drove Austria-Hungary to look southeast. The Seven Weeks War had established the North German Confederation, which would eventually become the German Empire, and had also seen a significant Italian advance into Veneto - an advance that did not go far enough in the eyes of Italian nationalists, who know looked longlingly at South Tyrol, Istria and Dalmatia, all still in Habsburg hands. In hindsight, it likely would have been better for Austria-Hungary to hunker down rather thank taking the offensive in Bosnia 1908 and then Serbia 1914, but that was simply not the diplomatic way of thinking back then.
The Balkan peoples wanted to push out the Great Powers, but different GPs depending on which country you're talking about. It is no isolated incident that a pan-nationalistic Serb killed the prince of a great power to cause World War One. The Serbians really didn't want the Austrians in Bosnia, where many Serbians lived - and definitely not on their doorstep at all. Serbia had witnessed Austria's reach for Bosnia in 1908 and was concerned about its own position - and rightly so, as proven by the fact that Austria's attack on Serbia was what kicked off the Great War on 28 July 1914. The Romanians wanted to liberate fellow ethnic Romanians living under the Hungarian and thus Habsburg yoke. Montenegro didn't have great plans of expansion, but they also didn't like the idea of the Habsburg kicking in their front door. Greece was still too small for its own liking and tried to take a bite from the Ottomans - which it eventually did in the 1912 Balkan War and then, as dessert, they even got some of the Bulgarian spoils in 1913 in round two.
The Russians wanted allies and economic influence. Russia was kind of the wildcard in the Balkans, as they didn't want to directly control the territory. That and their ethnic Slavicness made them appear as natural allies to several countries in the Balkans, but they eventually only ended up allying Serbia. Romania had preferred to enter into an assocation with the Central Powers to bide its time and see the diplomatic situation develop, Bulgaria viewed Russia as a possible friend but they absolutely hated the Serbians post 1913 and so eventually chose to cut their losses and end the rivalry with the Turks to over the Turks get on the good side of the Germans and eventually the Austrians to then in turn get a second shot at reclaiming the Balkan empire they had failed to conquer in the 1913 Second Balkan War. The Russians also were the country perhaps even more than the others concerned with its own prestige after losing a war to Japan of all countries in 1904. Foreign Minister Sergey Sazonov, one of the leading actors behind the tsar's throne, warned that Russia was at risk of becoming a second-grade power if after Japan, they also failed to repel Austria. Russia's main fixation in terms of its own wargoals were the Turkish-held Dardanelles, Russia's economic lifeline, as the Black Sea at that point was Russia's only viable and reliable access to international shipping waters, perhaps next to Baltic ports in Latvia and Lithuania - that were also not reliable since Denmark and especially Germany or the UK could cut off Russian supplies over the Baltics without much effort.
So, yeah. Plenty of potential for conflict even when Bismarck was still alive, a potential that then increased drastically after 1904 (Russia vs Japan), 1908 (Bosnian Crisis and Bulgarian Independence), 1911 (Italy vs Turkey), 1912 (First Balkan War) and 1913 (Second Balkan War).
Interesting question. While we wait for someone to take on the task of plumbing its depths, you may find the discussion in these past questions helpful:
How did the Balkans become the region of Europe most prone to ethnic conflict? Did either the Austro-Hungarian Empire (and predecessor states) or the Ottoman Empire have significant influence in this?. /u/commiespaceinvader's answer isn't hugely focused on WWI in particular, but is incredibly in depth and relevant.
To what extent did ethnic and nationalist tensions in the Balkans contribute to the outbreak of WWI?