We need to remove the envelope
so we can modify the fills. Select the enveloped checkerboard, then select Convert to Editable Shapes from the Arrange menu.
Ungroup (Ctrl - U) the checkerboard twice.
Select the large square and apply a Linear Fill using the two shades of black as shown (10-40%).
Select the checkers and apply a Linear Fill using the two shades of black as shown (100-70%).
Making the light and dark colors closer in value as they go to the back, enhances the feeling of depth.