Because 3 of those empires were tottering in the first place and had significant problems with minorities in the age of nationalism (every nationality wanted their own state when empires are explicitly based on imposing one state over multiple nationalities). The war was largely the thing which pushed them over the edge. I would go as far as to say the future of at least 2 of those empires (Austria-Hungary and Ottoman) were pretty bleak and likely to be in the danger of collapse even without a war.

In the Russian Empire for instance only around 45% of the population was actually ethnically Russian, in the Austria-Hungary empire only around 43% of the population was either German or Hungarian. The Ottomans Empire also had large parts of their territories populated by Arabs, Armenians, Greeks, Slavs, and other nationalities.

There was also deep political crisis in those empires, in Russia you had the 1905 revolution and continued resentment towards Tsarism and members of the Tsar's court (i.e Rasputin). In the Ottoman Empire you had fundamental question over how to deal with the nationalities, the loss of territories in the Balkan Wars and Libya to Italy, and the decline of the empire in general leading to the popularity of ultra-nationalist movements like the young Turks. In Austria-Hungary you had similar issue of how to deal with a situation where the German (and Magyar) elite ruled over a (largely Slav) rest of the empire. Basically in at least the Ottoman Empire the process of disintegration (in terms of territorial loss) has -already- started before the war.

WWI, as the first modern total war, placed unbearable strains on the physical, psychological and political structures of those empires. In the end, the various nationalities were enabled by the defeat of those empires in the war to gain (or at least attempt to gain) independence with the support of victorious powers (see treaties of Brest-Litvosky, Severes and Trianon). Those 3 empires collapsed both internally and externally.

Note I excluded Germany because it doesn't fall under quite the same category as the other three, while there was a Polish nationalities issue, it was small compare to the issue in the other 3. The actual level of political institutions and development was also much higher in Germany. The German Empire "fell" because the Kaiser (correction here!) forced to abdicate, it was not a collapse situation as with the other 3 empires.

But that being said, 3 of those 4 empires reconstituted themselves remarkably quickly (Germany with the Weimar Republic and later Third Reich, Ottoman with the founding of the Turkish Republic which fought to overturned much of the Severes treaty, and Russia with the victory of the Bolsheviks in reclaiming most of the ex-Tsarist Empire and founding the Soviet Union).

Source: War of the World by Nail Ferguson