There are a number of ways of solving this problem. One method is to count the unshaded squares in the diagram. There are $4$ complete squares, $8$ half squares and $4$ quarter squares, which is a total of $4 \times 1 + 8 \times \frac{1}{2} + 4 \times \frac{1}{4} = 4 + 4 + 1 = 9$.
Therefore there are nine shaded squares and nine unshaded squares, so half the area is shaded.
Alternatively, consider dividing the square up into the smaller red squares, shown in the diagram to the right. Each of the red squares is divided into two halves, one of which is shaded.
This means that half of the complete shape is shaded.