How does one interpret negative geopotential height values?
This likely means that the surface MSLP is less than 1000 mb.

If you are looking at a 1000 mb geopotential height, and the MSLP at a point is 995 mb, then the height of the 1000 mb geopotential surface would be below the ground, i.e. negative height.

Effectively the question was what does a negative difference of distance from a pressure surface mean? The answer is that the surface must lie below the frame of reference. In the example, the frame of reference is the ground and the surface is the 1000 mb surface.