tl;dr - Austria-Hungary had already gained de facto control of the region after the region 20 years before, and was simply taking advantage of Turkish turmoil to solidify their holdings. Russia, who had been backing Slavs in the region, including Serbia, was opposed to Austro-Hungarian ambitions. At the same time, Serbia had been rapidly expanding their sphere of influence and too saw Austria-Hungary as a threat. The annexation provided the perfect opportunity for Serbia to strike against their current greatest enemy.

The Annexation of Bosnia and Herzegovina by the Austro-Hungarian Empire essentially involved four different powers: the Austro-Hungarian Empire, the Russian Empire, the Balkans, and the Ottoman Empire. All four would prove essential in straining the European tensions leading up to the Great War, despite the fact that countless other factors were simultaneously at play.

Austria and the Ottoman Empire were both large, multiethnic conglomerates with powerful ruling families, but despite their similarities had both been in a longstanding struggle for dominance in Southeastern Europe since the middle ages. While the Ottoman Empire had generally held the upper hand prior to the Great Turkish War of 1699, following the Holy League's victory and Austria's acquisition of Hungary, the newly-formed Austro-Hungarian Empire would grow in power while their Eastern neighbors would continue to wane. Austria's incorporation of Hungary would set the stage for a political stalemate in the region however, as the fledgling Empire was now brought into the sphere of influence of Russia.

Russia, another old enemy of the Turks, recognized the Austro-Hungarian ambitions in the Balkans where the three empires now converged, but was much more willing and able to cooperate with the ladder due to a much longer history of aggression with the former. As such, Russia committed its resources in the Balkans to weakening local Ottoman authority by supporting the uprisings of various Slavic ethnic groups. Thus, Russian intervention strengthened and galvanized, among a plethora of others, the Serbian national identity.

The beginning of the political destabilization that would eventually lead to the Great War had its genesis in the wake of the crushing defeat of the Ottomans in the Russo-Turkish War and subsequent Treaty of Berlin in the Summer of1878. The treaty's goal was to ensure the survival of the Concert of Europe, a fragile informal agreement to conserve the 18th-century European political field. Because of this, Russia was notably forced to return the immense amount of territory claimed from the preceding Treaty of San Stefano. However, the reversal of Russian territorial claims didn't remedy the irreparable damage done to Turkish influence in the Balkans. For example, Bulgaria, although officially still a part of the Ottoman Empire, would act as a sovereign nation and Serbia, previously operating as a de facto free state, would officially gain its independence.

The most influential result of the treaty, however, was that Bosnia and Herzegovina would be nominally returned to the Ottomans, but in reality administrated by Austria-Hungary. The intention was that Austria-Hungary would only temporarily govern Bosnia and Herzegovina, but as the time given for the administration was indefinite, Austria-Hungary had no reason to give up their only recent major geopolitical victory.

Back to the overall picture in the Balkans, the Austro-Hungarian and Ottoman Empires were no longer competing directly with one another, but now a variety of smaller Russian-backed Slavic states, with the Austro-Bulgarian political conflict being the most influential. Thus, Austro-Russian Relations deteriorated as the former attempted to expand its influence, while at the same time Austro-Turkish relations cooled.

Thus, when Austria-Hungary officially annexed Bosnia and Herzegovina during the Young Turks' coup of 1908, it had little effect on the Ottoman Empire, which, aside from losing actual control of the territories 20 years ago, was in turmoil due to the coup and was simultaneously having to deal with the Bulgaria's Declaration of Independence. Rather, it frustrated Russia and other Great Powers who wanted to preserve the status quo that Austria-Hungary had destroyed.

Serbia was indeed disheartened by the official loss of Bosnia and Herzegovina, as the territory not only housed many Serbs and had a shared history and culture with the Kingdom, but also served as a geographical obstacle between them and their close ally Montenegro. However, not immediately affected by the annexation, Serbia simply redirected their attention to Turkey. They defeated the Ottomans and subsequently Bulgaria in the Balkan Wars, gaining significant territory in the process. However, Austria-Hungary would still prove to be a deterrent to their ultimate goal of freeing and uniting all Serbs in their irredentist campaign, and thus the shadowy Black Hand organization, which had countless rebel cells in all areas of the Balkans, orchestrated the assassination of Austrian Archduke Franz Ferdinand by making use the Bosnian opposition to the Habsburg regime.

So, the First World War wasn't necessarily triggered by Austria-Hungary's annexation of Bosnia-Herzegovina, but it did provide Serbia with the means to strike against them, catalyzing the road to war. Serbia had strong, even imperialist ambitions in the Balkans, so its unlikely the prevention of the annexation alone would've prevented conflict between it and Austria-Hungary.