Explanation

We see the 8 x 12 rectangular box as shown in the figure.
A single rectangular insert is folded twice to make four identical compartments. We see in the figure that we have 4 identical right triangular compartments. So the insert is just the lightning shape shown in the figure. We need its area. It is actually a rectangle that is folded into this shape. If we open it up, we will get a rectangle and its area will be the total height * Width (which is 8 since the box has a square base of side 8)

Now, all we need is the height. Since the compartments are equal, the height of each compartment is 6. So each diagonal in the figure is 10 (using multiple of Pythagorean triplet 3-4-5).
Total height of the rectangular insert = 10 + 8 + 10 = 28

Its area = 28*8 = 224