Example 1 Let $f: \mathbf{R} \rightarrow \mathbf{R}$ satisfy
$$
f\left(x^{3}\right)+f\left(y^{3}\right)=(x+y) f\left(x^{2}\right)+f\left(y^{2}\right)-f(x y) \text {. }
$$

Find the analytical expression of the function $f(x)$.
$(2012$, Albanian Mathematical Olympiad)
【Analysis】(1) Observing that $x$ and $y$ are relatively scattered, one can remove a variable by assigning one of them to 0, thereby observing the other variable. Based on this, finding the value of $f(0)$ becomes the starting point of this problem.
(2) Let $x=y=0$. Then
the left side of equation (1) $=2 f(0)=0=$ right side.
Therefore, $f(0)=0$.
According to the analysis in (1), let $x=0$. Then
$$
f\left(y^{3}\right)=f\left(y^{2}\right) \text {. }
$$
(3) (4) Substitute equation (2) into the original equation (1), and let $x=y$. Then $2 f\left(x^{3}\right)=2 x f\left(x^{2}\right)$.
Substitute equation (2) to get $(2 x-2) f\left(x^{2}\right)=0$.
When $x \neq 1$, $f\left(x^{2}\right)=0$.
Let $x=1, y=2$, substitute into equation (1) to get
$$
f(1)+f(8)=3 f(1)+f(4)-f(2) \text {. }
$$

From $f\left(x^{2}\right)=0$, we know
$f(2)=f\left((\sqrt{2})^{2}\right)=0$,
$f(4)=f\left(2^{2}\right)=0$,
$f(8)=f\left((2 \sqrt{2})^{2}\right)=0$.
Thus, $f(1)=3 f(1) \Rightarrow f(1)=0$.
(5) $f\left(x^{2}\right)=0(x \in \mathbf{R})$.

From equation (2), we know
$$
\begin{array}{l}
f\left(x^{3}\right)=f\left(x^{2}\right) \Rightarrow f\left(x^{3}\right)=0(x \in \mathbf{R}) \\
\Rightarrow f(x)=f\left((\sqrt[3]{x})^{3}\right)=0(x \in \mathbf{R})
\end{array}
$$
$\Rightarrow f(x)=0$ is the unique solution to the original equation.
Observing the internal connections in the problem, such as the scattered nature of $x$ and $y$ in this problem, plays a positive role in solving the problem.