Example 1 Given the sets $P=\left\{m^{2}-4, m+1,-3\right\}, Q=\{m-3,2 m-1,3 m+1\}$, if $P \cap Q=$ $\{-3\}$, find the value of the real number $m$.

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Analysis We use the intersection $\{-3\}$ as the entry point.
Solution
(1) When $m-3=-3$, we get $m=0$, so $P=\{-4,1,-3\}, Q=\{-3,-1,1\}$, and thus $P \cap Q=\{1,-3\}$, which does not meet the requirement.
(2) When $2 m-1=-3$, we get $m=-1$, so $P=\{-3,0,-3\}$, which does not conform to the distinctness of set elements.
(3) When $3 m+1=-3$, we get $m=-\frac{4}{3}$, so $P=\left\{-\frac{20}{9},-\frac{1}{3},-3\right\}, Q=\left\{-\frac{13}{3},-\frac{11}{3},-3\right\}, P \cap Q=\{-3\}$.
Therefore, the solution to this problem is $m=-\frac{4}{3}$.
Comment This example, based on the definiteness and unordered nature of elements, provides a classification and listing, and then checks the results according to the distinctness of elements.