Example 8 Solve the equation $3 x^{3}-[x]=3$.

Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.
Solution: For higher-degree equations containing $[x]$, discussing by intervals can be an effective method.
If $x \leqslant -1$, then $3 x^{3} - [x] \leqslant 3 x - x + 13 x - x = 2 x \geqslant 4$.
If $0 \leqslant x < 1$, then $3 x^{3} - [x] = 3 x^{3} < 3$.
When $1 \leqslant x < 2$, $3 x^{3} - [x] = 3 x^{3} - 1$. The original equation becomes $3 x^{3} - 1 = 3$. Therefore, the solution to the equation is $x = \sqrt[3]{\frac{4}{3}}$.