1.  Since the volume of the small cube is 8 cubic inches, the size of its edge is 2 inches  (2^3 = 8).  

    Do you understand it ?



2.  Since the bottom of the box is 9 inches X 4 inches,  you can place 4 small cbes in one direction (2*4 = 8)  
    and  2 small cubes in the other direction  (2*2 = 4).

    So, you can place exactly 8*2 = 16 small cubes in one layer at the bottom of your box, and you can not place more into this layer.

    Do you understand it ?



3.  You can place all yours 24 small cubes in three such layers into the shipping box.

    Do you understand it ?



4.  Hence, the minimum height of the shipping box is 3 times 2 inches, which is 6 inches.