Let us consider the side of the big square as a
The area of the big square would then be $a^2$

Any diagonal cuts through the square in equal halves. With 2 diagonals the square is split in 4 equal sizes.
The area of the big triangle consisting the shaded square is $\frac{1}{4}$ $a^2$

If you take any diagonal of the shaded square its length would be $\frac{1}{2}$ a
The shaded square can be divided into two equal triangle of 45-45-90
The length of such a triangle is $x:x:x\sqrt{2}$
Therefore, $x\sqrt{2}$ = $\frac{1}{2}$ a
From this we can calculate a side of the shaded square as $1/2\sqrt{2}$*$a$
Area of the shaded square would be $\frac{1}{2}\sqrt{2} * a$* $1/2\sqrt{2} * a$ = $\frac{1}{8}$ * $a^2$
Hence, this enables us to calculate the percentage as:
$\frac{1}{8}$ $a^2$ / $a^2$ * 100
$a^2$ cancels out giving $\frac{100}{8}$ = 12.5 %