The full phrase was something along the lines of "Italian master race destined to rule over and civilize all lesser groups". It was made in the context of the panic Romans had with regards to the Germanic tribesmen which contested that "master race" perception. In full when, if ever, did the Romans/Latins start to view themselves as a "master race" and did the Germanic tribesmen ever manage to shake certainty in this mindset.
It sounds like he's referring to some famous passages from Vergil's Aeneid along those lines. From Fitzgerald's translation (Jupiter is speaking) :
"[…] young Romulus
Will take the leadership, build walls of Mars,
And call by his own name his people Romans.
For these I set no limits, world or time,
But make the gift of empire without end."
and another famous passage (again, Fitzgerald's translation):
"Roman, remember by your strength to rule
Earth's peoples—for your arts are to be these:
To pacify, to impose the rule of law,
To spare the conquered, battle down the proud."
So by Augustus' reign, when Vergil was writing, there was at least some perception that the Romans were meant to rule other peoples, to pacify them, and to bring them the 'rule of law'. How much that was general and how much was Vergil writing for the elite, I leave to others more knowledgable to answer. I'll also leave it to others to answer your question w/r/t to the Germans.
Wait the rhine has video? I'm a huge fan of that show.