Example 1 Let the set $A=\left\{a^{2}, a+1,-3\right\}, B=\left\{a-3,2 a-1, a^{2}+1\right\}$, and $A \cap$ $B=\{-3\}$, find the value of the real number $a$.
Since $A \cap B=\{-3\}$, we have $a-3=-3$ or $2a-1=-3$, which means $a=0$ or $a=-1$.
When $a=0$, $A=\{0,1,-3\}, B=\{-1,1,-3\}, A \cap B=\{1,-3\}$, which contradicts the given information, so we discard it.
Therefore, $a=-1$.