Won't do a long writeup, i'm not an expert in Roman history or Fascism in particular, but I believe it's more of a connection with Fascism in general than just the Nazi's in particular. Mussolini was in love with the Roman Empire, and basically modeled most of Fascism behind it, using the Fasces as his symbol, a symbol previously used by the Romans as well. (It is also the root of the word Fascist). The Grandeur and glory of the Roman Empire served as an inspiration for countless empires throughout history, and the Fascists were no exception. The structure and authoritarian nature of a lot of the more militaristic aspects of Roman society was highly appealing to Fascists, they demanded professionalism and structure. The Germans liked this, in the Interwar years, German generals like General Hans von Seeckt, were rather happy about the restrictions posed on the Treaty of Versailles, he saw the limitation of the army to 100,000 men as a way to ruthlessly purge the Army of inept officers, and as a perfect basis to create an elite and proffesional army. German proffesionalism and discipline stemmed partly from its Prussian roots, and when the Nazis took over in 1933, bringing their Romanized authoritarian ideology with them, Roman attitudes seemed to blend fairly well with the already disciplined and well organized Military.

Someone with more knowledge should post a better answer than mine though :)

In a way Nazism was the antithesis of the Roman Culture. Nazism was based on a racist worldview, where your race determined your worth and status. Because of this, no non-German people could become Germans if they were not of the German Race. Rome on the other hand considered people Roman if they assimilated into Roman culture and acquired citizenship, and because of this many Roman Emperors came from previously conquered peoples. So to say that Roman culture influenced Nazism would be wrong. Maybe the Nazism borrowed some cultural symbols, but they adopted a worldview that totally broke with what the Roman culture represented.

Association with the Roman Empire brought prestige and their symbols and ideals have been emulated throughout history not just by the Fascists. Think of the Holy Roman Empire which was non of what the name suggests, our financial and law buildings which emulate Roman austerity, and the eagle being used by many organizations. Roman culture brought a sense of legitimacy and a notion to states thus states have been trying to copy their glory since their collapse.

It was not uncommon at all in Western Europe for kings and generals to compare themselves and their civilizations to Rome or Greece. You must remember that Greek and Roman culture were emulated strongly throughout all of Europe, especially during the Aristotalian movement and the Renaissance. Because of this, we find historians in the 15th and 16th century coming up with the idea that the millennium from 500 C.E. to 1500 C.E. was a "dark age."

This classification and abandoning of recent history led the aristocracy to look to the Romans and Greeks for inspiration: Alexander the Great and Julius Caesar, Augustus and Diocletian. The Romans and Greeks were considered superior to those that came before and after them.

Now we bring in the Nazis. This political party, and eventually the state of Germany, came to believe that it and its people were superior to all else. They planned to take over great swathes of land and create an empire, the 3rd Reich. To show this superiority and dominance, the Nazis adopted certain customs that belonged to the only empires "worthy" of emulation: Greece and Rome.

I am by no means an expert on 20th century facsim, however. I just thought this was an ok explanation. Feel free to correct me.