Well the Balrog is Not a mindless creature. He is a fellow Maiar Like Gandalf himself. Sure He got twisted horribly and no longer resembles what He was. But still.

I once read an explanation of what the Flame of Arnor and Udum means but it was rather complicated and i do Not remember.

But i do think it Had the purpose of scaring him. Gandalfs Power is limited in His current Form and He knows He cannot Beat the Balrog by Sheer Power.

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https://castellan-of-the-ring.blogspot.com/

Like Asgarod said, the Balrog was able was probably able to understand Gandalf despite its long isolation/imprisonment. I think Gandalf saying he was a servant of the secret fire, wielder of the flame of Arnor was a script error. In the books, Gandalf says he is a wielder of the flame of Arda, which is the world. In the Silmarillion's first section, where the world is made, the planet is built around a flame called the flame of Arda and the secret fire. So what Gandalf is basically doing is declaring to the Balrog that he is a Maiar that is still on side with the "old boss" and is not to be trifled with.

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"Draw your sword with a heavy heart, but swing it with a heavy hand"

In one of Tolkien's letters, he likened the secret fire to the holy spirit. But as Wan said, its saying the same thing. I am on the side of the one.

He's not saying Arnor, he's saying Anor which is the Sindarin name for the Sun. The "flame of Anor" would be the light of the Sun, which originated in the fiery fruit of Laurelin, one of the Two Trees of Valinor. It may also refer to Gandalf's possession of Narya

Well, that doesn't make sense to me, but then, the internet does seem to think it is correct. Learned something new there.

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"Draw your sword with a heavy heart, but swing it with a heavy hand"