The shape can be folded like this
Then it should become obvious.
Method of solving:
I started out by cutting out the shape, noticed some equal lengths, and initially folded it in the wrong places (as can be detected by creases in my model). I then wondered if the four smaller rectangles would wrap around four sides of the cube. I marked the parallel diagonals of the four small rectangles, and noticed that a line touching the tips of their saw-tooth edges is at 90° to them. I could then mark out the two other faces in the largest squares, and from there it all fell into place, leaving the larger squares as the top and bottom caps, folded to fill in the gaps.
As I was working with a print of the black image in the question, it proved rather difficult getting a photogenic shot, and I ended up retouching it to show where the original edges are.
