Imagine how Britain would react today if a nationalist from another country had murdered Crown Prince Charles. Archduke Franz Ferdinand was next in line for the throne of Austria-Hungary, and monarchies in those days took the line of succession very seriously. It was not considered merely murder, but an attack on the empire itself.
They responded by invading Serbia with tens of thousands of troops. Russia responded in kind on behalf of their Serbian ally, and all of the European empires were at war within a week.
The quick answer to this is that they went to war. That is not exactly true, but it's close enough.
More technically, what they did was to issue an ultimatum to the government of Serbia. They essentially told Serbia "If you do not do this list of things, we will go to war with you."
Because the Austro-Hungarians pretty much wanted a war with Serbia, they made the conditions so harsh that there was no way the Serbs would agree to them. When Serbia did not agree, Austria-Hungary went to war.