Side length of original cube was 1000 mm
Just one small cube has of edge length 1 mm.
Comparison of areas goes as square of side length ratios.
Surface area for one small cube only is 1/1,000,000 of original.
There will now be 1000,000,000 small cubes, each of 1/1,000,000 of original.
That explains why the factor for total surface area increase is 1000