To my understanding, Austria has never been ruled by Germany or any German power, and has a history of political independence and achievement on its own. So why does it speak German? Did Austria have close historical ties with Germany?
The territory that is now Austria has been German speaking ever since there has been a German language. The German language predates any modern polity, it evolved from older Germanic languages during the Great Migration period. Austria was controlled by a tribe known as the Bavarii, who displaced and mingled with Slavic and Rhaeto-Romansch groups in the area, forming a culture that spoke a proto-Germanic language.
There didn't used to be "the Germany". The region around (today's) Germany and Austria consisted of several German-speaking nations who were all Semi-Allies (They were all part of a greater Empire, but they still went to war with each other)
As time progressed, 2 of these German nations became very dominant: Austria (in the south) and Prussia(in the north). The final war between the two was won by Prussia. Prussia decided that they wanted to shape the fate of Germany themselves, and since Austria was too powerful, they had to leave. So one of their peace conditions was that Austria would go separate ways from then on.
After that event, "The" Germany (German Kaiserreich) was actually created and the King of Prussia became the Kaiser of the German Kaiserreich, which was the Kaiser (Emperor) of all the German-Nations (except Austria).
There has always been talks of "Großdeutschland" (big Germany) which consists of Germany AND Austria. And the both nations always used to be close allies after that, because they were both German-speaking.
As daedalus mentioned, Austria was settled by the Bavarii, who also settled the area in Germany we know as Bavaria (in Southern Germany). Austria is derived from the German word "Osterreich," or Eastern (part) of the Empire. This is relating to the Holy Roman Empire that was the primary political entity in Germany at the time. For a long time, the Austrians were just another part of the Germanies, as were the Saxons, North Germans, and so forth.
The reason why Austria was able to retain political independence was partly due to the Habsburg dynasty, who were able to gain power as the Holy Roman Emperor and subjugated nearby territories, including Hungary, Bohemia, and parts of the Balkans. As a result, when the Holy Roman Empire fell in 1806, Austria had already created a large empire that was independent from the other German states that unified under Prussia. And thus it remained politically independent from the rest of the German Empire.