The first shaded triangle has legs $3$ and $3$, so its area is $\dfrac12\cdot3\cdot3=\dfrac92$.

At each later step, the relevant triangle has half the leg length, so the shaded area is multiplied by $\left(\dfrac12\right)^2=\dfrac14$.

The total shaded area after many steps is therefore very close to the geometric sum $$\begin{gathered} \frac92\left(1+\frac14+\frac1{16}+\cdots\right) \\ {}= \frac92\cdot\frac1{1-1/4} \\ {}= 6. \end{gathered}$$

After $100$ steps the omitted tail is tiny, so the total area is nearest $6$.

Thus, A is the correct answer.