In short, they didn't. 99% of voters, if we trust the fairness of Nazi elections, voted in favor of being annexed.
But let's dig a little deeper and take a look at some reasons as to WHY Austrians and Germans supported the annexation of Austria in 1938. First of all, even though it might be hard to find an Austrian who doesn't get offended by being called German, this is primarily a postwar phenomenon. Germans and Austrians have been united for far longer than they haven't been. Be it through language (German, yes, but specifically the various Bavarian dialects), political unity (Germany and Austria were both part of the Holy Roman Empire from 962-1806), or culture (even today, you'll find a lot more differences between Munich and Berlin than you will between Munich and Salzburg), Germany and Austria have historically been very close and sometimes even united.
Furthermore, it is necessary to look at the question from the context of time. Because of the Great Depression, economies around the world went into shambles, and that of Austria was no exception. Now, it is important to note here that Austria was in no way, shape, or form used to being in the position it found itself. Since 962, Austria had been an integral part of powerful empires - first the Holy Roman Empire (until 1806), then the Austrian Empire (1804-1867), then, finally, the Austro-Hungarian Empire (1867-1918). After almost a millennium of being a dominant force, Austria fell far.
Then, after a mere 20 years, here comes Hitler, who the Austrians have clearly seen has made great strides in Germany, increasing its stature in both military power and economically in just a few short years. Austrians in 1938 often felt just as cheated by their loss in WWI (both militarily and territorially) as did Germans. For many, the appeal to regain what was lost was great and it wasn't lost on them that the leader of Germany, Hitler, was, in fact, Austrian himself.
A quick note about antisemitism: Antisemitism was common throughout Europe (and America). The fact that Jews were being persecuted in Germany meant little to a country that was actively prosecuting Jews without the Germans' help. In fact, the DNSAP (Deutsche Nationalesozialistische Arbeiterpartei) was around from 1918 until its merger with the NSDAP (Nationalsozialistische Deutsche Arbeiterpartei) in 1933. Austrians - indeed no one - required German assistance in persecuting Jews. The primary reasons for supporting annexation were cultural and economic. Of course, Hitler's calls to rid Europe of Jewish people did little to nothing to cause Austrians to question if annexation was the right way to go.
