Step 1: State the goal cleanly.
We want the total volume of the spheres that actually live inside one body centred cubic unit cell, written using the edge $a$.
Step 2: Recall the count of atoms in BCC.
A BCC cell owns $8$ corner atoms shared as $\tfrac{1}{8}$ each plus one whole body atom, so $Z = 8\times\tfrac18 + 1 = 2$.
Step 3: Relate radius to edge using the body diagonal.
In BCC the atoms touch along the body diagonal of length $\sqrt{3}\,a$, which equals four radii, so $\sqrt{3}\,a = 4r$ giving $r = \dfrac{\sqrt{3}\,a}{4}$.
Step 4: Write the occupied volume as a product.
Occupied volume $= Z \times \dfrac{4}{3}\pi r^3 = 2 \times \dfrac{4}{3}\pi r^3$.
Step 5: Substitute the radius.
\[ 2 \times \frac{4}{3}\pi \left(\frac{\sqrt{3}\,a}{4}\right)^3 = 2 \times \frac{4}{3}\pi \cdot \frac{3\sqrt{3}\,a^3}{64} \]
Step 6: Simplify to the final form.
\[ = 2 \times \frac{\sqrt{3}\,\pi a^3}{16} = \frac{\sqrt{3}\,\pi a^3}{8} \]
So the spheres fill $\dfrac{\sqrt{3}\,\pi a^3}{8}$ of the cell, option (A).
\[ \boxed{\dfrac{\sqrt{3}\,\pi a^3}{8}\ \text{(option A)}} \]