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I think from an inside perspective to the story there's really no telling what the curses were all about. Except perhaps that they are connected to the Great Music - in the Elder Days more accessible than later on one might assume - and that characters in a moment of epiphany might get a clear vision of the (always so ironic) future through a glimpse into the Great Music. Such as when Glorfindel prophets the end of the Witch-King. This fore-seeing is mostly associated with Elves but one might assume that some mortals also have this latent ability to hear the Music so to speak.

But really, I think these curses and prophesies (that always come true) would be better understood from an outside perspective. After all, Tolkien's inspiration for these stories - the Nordic, Old English, Classic and I don't know what mythologies - are ripe with these kinds of things, and that they should pop up in his own books is understandable, no? I suppose they also help to give the books that flavour of Old Tradition Tolkien is looking for.