It's another way to dictate the string-height at the 1st fret, as the "zero"-fret makes that the same as compared to another fret with the strings barred at the preceding fret. And with a zero-fret, the nut acts more as a string spacing guide and less, if at all, as setting the string height at that end. And, as Bill states, it gives open strings about the same sound as fretted ones.
On a standard design guitar, the nut is the beginning point for the scale-length of the strings; with a zero-fret, the zero-fret is that beginning point, and the nut is located behind that point.
Both zero-fret and standard nut/1st-fret designs need to be crafted and set-up with precision, although the nut behind a zero-fret is a little less critical in some respects.
Note that a lot of vintage Gretsches came with zero frets, just one example of many.
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