Part A
See figure 1 in the attached images below. We are simply reflecting the octagon over the vertical line L. Octagon A is the original octagon and octagon B is the reflection. I'm using A instead of "O" because this letter is easily mixed up with the number zero.
-------------------------------------------
Part B
See figure 2.
Reflect octagon A over line M and you'll get octagon C.
Reflect octagon A over L first, then over M next, and you'll get octagon D. A simpler way to think of getting octagon D is to reflect octagon B over line M.
-------------------------------------------
Part C
If we have n sides of a regular polygon, then E = 360/n is the exterior angle measure. In this case, n = 8, so E = 360/n = 360/8 = 45 degrees is the exterior angle measure. This is the blue angle as shown in figure 3.
Doing a reflection will double the angle to 45*2 = 90 degrees which is one of the interior angles of the quadrilateral. This applies to the other 3 interior angles as well (since we have a bunch of reflections). So far we have proven the inner quadrilateral is a rectangle
The quadrilateral has all 4 sides the same length (due to the octagon's being reflected, so the corresponding pieces have the same length), so we have a rhombus.
A quadrilateral that is both a rhombus and a rectangle is best known as a square. If we had a Venn Diagram of the set of all rhombuses and the set of all rectangles, then the square of all squares would be on the overlapping portion of the two sets.
