How did Hitler justify that the Austrian people were their German brethren?
You appear to be mixing up various stages of history here.
The idea of a nation state is relatively new; historically, most entities that we consider states may have had one defining ethnic or cultural group, but by today’s standards, we should consider them multiethnic states. That’s certainly true for Germany which was a chaotic pattern of rather small states, bound in the Holy Roman Empire of German Nation which existed until Napoleon’s conquests in 1801. That first German empire consisted of what we know today as Germany but also of Austria. So, historically, Austria used to be one of the German states. After Napoleon, the German Federation was established, a weird construction causing bits of Prussia to be inside Germany and other bits outside of it, and the very same happened to Austria. Still: Even after 1815, Austria continued to be a part of Germany.
Austria, however, had pursued a policy of strategic marrying for several centuries – dubbed popularly as »Bella gerant alii, tu, felix Austria, nube!« (Literally: May the others go to war, you, lucky Austria, marry). By doing so, Austria had overtaken power in what is today the Czech Republic, Slovakia, Hungary as well as parts of other modern countries (clockwise from the East: Poland, Ukraine, Romania, Serbia, Croatia, Slovenia, Italy). This was considered Austria.
In 1848, Germans started a revolution in the pursuit of a nation state. And Austria – still a part of Germany at this time – happened to be a problem:
Consequently, the 1848 revolution failed. The status quo ante was kept: German Federation with Austria being a part of it.
In 1867, Austria re-modelled its structure, forming a bipolar structure with Austria and Cisleithania on the one hand and Hungary or Transleithania on the other. This is what we know as Austria-Hungary today. And its Austrian part still remained a member of the German Federation.
Only four years later, in 1871, Prussia unified (Lesser) Germany. The German Empire – technically the second after the Holy Roman Empire of German Nation – now excluded Austria for the first time in hundreds of years.
So contrary to what your question sounds like, the Germany Empire and Austria-Hungary existed next to one another only between 1871 and 1918. That's less than half a century in contrast to the rest of the second millennium during which Austria was considered one of the German states. So we have the same language, a gradually shifting culture (meaning that the culture of Bavaria is considerably closer to Austrian culture than it is to what you see around Hamburg or other North German areas), and hundreds of years of shared history – one really shouldn’t wonder why strong ties were felt.
Austria lost its possessions after the Treaty of Saint Germain. This finally solved the problem of 1848: Austria, stripped off its non-German-speaking population could have merged with Germany to form the German nation state many people had been dreaming of throughout the 19th century. But that was politically illegal: The peace treaties of Saint Germain (for Austria) and Versailles (for Germany) both made it strictly illegal for Austria to join Germany and form Greater Germany. When suddenly the solution of this old problem seemed so feasible, this was seen as one more of the injustices of the victorious powers of World War I.
Hitler at last, an Austrian who had lived in Munich and fought for Germany in World War I, thus did not consider himself Austrian – as in being distinct from a (pan-)German identity. And since one of his major electoral promises was to get rid of all the limitations of the Versailles Treaty, the Anschluss, the annexation (or rather: unification) of Austria was considered the most logical of his political steps.