Generally speaking, people in Austria speak German, but it's not quite so straightforward as that. The reason why Austria doesn't have its “own” official language is because a large chunk of its history was tied up with Germany's (as in, it was part of Germany for some time).

Why is German the official language of Austria?

Some words come from Latin and Greek, while others are borrowed from French and Modern English. German became the 'official' language of Austria in the 18th century when the Roman Empress Maria Theresa and her son, Joseph II, introduced compulsory schooling in their multilingual Habsburg empire.

How did Austria speak German?

Although Croatian, Hungarian, Slovenian, Turkish, and other languages are spoken by the various minority groups, nearly all people in Austria speak German. The dialect of German spoken in Austria, except in the west, is Bavarian, sometimes called Austro-Bavarian. About seven million people speak Bavarian in Austria.