While large parts of what would become the Holy Roman Empire were not conquered by the (classic) Roman Empire, it wasn't an unconquered area. One important factor in the development for example was Charlemagne subdoing the Saxons. Charlemagne was reigning over a large part of Europe that included what became France and what became the Holy Roman Empire. Louis the Pious, son of Charlemagne, divided the empire under his three sons, although they were still connected in the sense, that one of the sons (Lothair I.) was put above the others. However, the development from there was going to a stronger separation. This is important, because you can see, that the area, you know as Holy Roman Empire, stands in tradition of Charlemagne. It is necessary to know this for understanding the main part of your question.

I will focus on the "Roman" in Holy Roman Empire, since i feel that this is the part you're most interested in. You may know, that the people of the middle ages had a strong religious feeling and believe. This is to keep in mind not only for peasants, but also for the savants and the elite. In the bible, book Daniel, it is said, that the Babylonian king Nebuchadnezzar II. had a dream of a statue, that was made of four different metals. Bothered by this dream, Nebuchadnezzar searched for an explanation, what the dream may mean. Daniel then explains him, that the four metals of the statue are a symbol for the four big empires, that will rule the world, before the world will end with the reign of god. The start for that would be the Babylonian Empire of Nebuchadnezzar himself.

So far for the bible. Later it was interpreted, that the four empires were the Babylonian Empire, the Median/Persian empire, the empire of Alexander the great and the Roman Empire. Now with the end of the domination of the Roman Empire, and the Empire itself, the world should have ended, if you take this interpretation literal. How to explain that it didn't end, that the world is still the same? It was drawn the conclusion, that the Roman Empire still existed. The explanation given is, that because Charlemagne was crowned as Emperor in Rome, the Roman Empire devolved to him. This idea is known as translatio imperii. And since the "German" area of his former Empire, stands in his tradition, it is a continuity of the Roman Empire, hence the name.