Solution:

We can find the area of the shaded part by subtracting the area of the unshaded part from the whole area.

There are $4$ squares with side length $1,$ contributing $1$ each to the unshaded area, for a total of $4.$

There are also $4$ triangles with base $3$ and height $\dfrac{5}{2}.$ This contributes a total area of $$4 \cdot \dfrac{1}{2} \cdot 3 \cdot \dfrac{5}{2} = 15.$$

The total unshaded area is therefore $4 + 15 = 19.$

The total area is $5^2 = 25,$ and subtracting the unshaded area yields $25 - 19 = 6$ as the area of the shaded region.

Thus, B is the correct answer.