How did the desire for national independence among ethnic groups weaken and ultimately destroy the austrian empire

Starting with the creation of the Austro-Hungarian Empire everything was practically doomed to collapse, really. The Hungarians were the only group truly recognized by Austria, but there were plenty others that wanted a say in the government that ruled over them, like the Serbs. Though Serbia became independent from Turkey in the 1890's, most Serbians actually lived in territories claimed by the Austrians! When the Serbs living in Bosnia realized that the Austrian government was not concerned with how they felt about things, they decided to create a fanatical underground resistance movement financed by the Serbian government which lasted several years. Finally, in 1914 a Serbian nationalist named Gavrilo Princip shot and killed the Austrian archduke, which started WW1. So the answer to your question is that the desire for national independence led to fanatical resistance movements which ultimately began World War One.
Hope I helped!