The mourning of the earth's kings is the whole apparatus of the lower nature grieving as the great outward city of illusion goes up in flame. Babylon is the mind of confusion, the built-up empire of mingled thoughts and sensual indulgence a person has lived inside, and the kings are the governing appetites that ruled the body and coupled themselves to her pleasures. Fornication here is the mixing of the pure awareness with material craving, the self selling its finer energy for the taste of the passing world.
What burns is the false structure of the sensual mind, and the powers that fed on it can only stand off and wail, watching the smoke rise, because they were never anything but her partners. The ruling desires lament because the source of their delight is consumed and there is nothing left of it but rising ash.
Kings of the earth: the governing appetites of the body that ruled and coupled with the sensual mind
Her: Babylon, the built-up city of confused thought and material craving inside a person
Smoke of her burning: the visible passing of the false sensual structure as it is consumed