What others have said about this song representing the broken lord of flames makes sense and all, and I think it does, but I think it goes deeper than that.  I think it represents you.  Not you as the chosen undead, you as the player.

You will never again experience Dark Souls again like you did the first time.  Sure, you can play it over again, but it doesn't feel the same when you know whats coming.  And that's kinda sad.

Never again will you feel the crap-your-pants terror when the Asylum Demon crashes down and all you have is a broken sword, and the sense of triumph when you come back to drop plunge on him for a huge chunk of HP.

Never again will your heart sink when the second gargoyle emerges, only to burst with pride when you defeat them both.

Never again will your jaw drop at the beauty of seeing Anor Londo for the first time.

Never again will you get that first adrenaline rush from fighting O&S, and holler out in sheer ecstasy when you finally manage to overcome them.

Never again will you feel those feels when Sif is limping around pitifully (well you might but it wont be like the first time).

Never again will you pass through the final boss fog, expecting a badass lord conqueror, only to find a broken man, with a depressing, somber soundtrack.

This song signifies your last time truly experiencing Dark Souls, and it really is quite sad. That's what I think the true meaning of this song choice was.  Maybe I'm wrong, but it makes sense to me.