Example 4. Fold a square $ABCD$ with side length 3, with the crease being $EF$ (as shown in the figure), so that point $B$ lands on point $B'$ on $CD$, the extensions of $BA$ and $FE$ intersect at point $A'$, and $\angle B'BC=30^{\circ}$. Try to calculate the area of $\triangle A'EG$.
Let $B B^{\prime}$ intersect $A^{\prime} F$ at point $O$. It is easy to prove that $\triangle B C B^{\prime} \cong \triangle A^{\prime} O B$, and $B B^{\prime} = 2 \sqrt{3}$.
$$
\begin{array}{l}
\quad \therefore A A^{\prime} = A^{\prime} B - A B = 2 \sqrt{3} - 3, \\
\angle A G A^{\prime} = 30^{\circ}. \\
\therefore A G = \sqrt{3} A A^{\prime} = 6 - 3 \sqrt{3}, \\
A E = \frac{1}{\sqrt{3}} A A^{\prime} = 2 - \sqrt{3}. \\
\therefore B G = 2(2 - \sqrt{3}).
\end{array}
$$

Thus, $S_{\triangle A^{\prime} E C} = 7 \sqrt{3} - 12$.