How did Serbia anger Austria-Hungary?
Archduke Franz Ferdinand was assassinated in Serbia by Gavrilio Princip.
Bosnia and Herzegovina was under Austria-Hungary's control that time. The Black Hand, a Serbian group, wanted to free Bosnia and Herzegovina because 80% of Bosnia and Herzegovina's citizens were Serbians.
The Black Hand planned to combine Serbia and Bosnia and Herzegovina to make a Greater Serbia. So the Black Hand planned to assassinate the Archduke when he was in Serbia.
Gavrilo Princip, a member of Black Hand, was the assassin who killed the Archduke (other assassins failed). The Archduke along with his wife died by a gunshot.
On July 28, 1914, Austria-Hungary declared war on Serbia.
I have answered a question that is somewhat similar to this before so here's the link if you want more details: