Example 8 Let $a, b, c \in \mathbf{R}^{+}$, satisfying $a+b+c=a b c$. Find the minimum value of $a^{7}(b c-1)+$ $b^{7}(a c-1)+c^{7}(a b-1)$.
Given that $a, b, c > 0$, and $a + b + c = abc$, we have $c(ab - 1) = a + b$.
Similarly, we get $b(ac - 1) = a + c$, and $a(bc - 1) = b + c$.
By the AM-GM inequality, we have
$$abc = a + b + c \geqslant 3 \sqrt[3]{abc},$$

which implies $abc \geqslant 3 \sqrt{3}$, with equality holding if and only if $a = b = c = \sqrt{3}$. Therefore,
$$\begin{aligned}
& a^{7}(bc - 1) + b^{7}(ac - 1) + c^{7}(ab - 1) \\
= & a^{6}(b + c) + b^{6}(a + c) + c^{6}(a + b) \\
\geqslant & 6 \sqrt[6]{a^{6} b a^{6} c b^{6} a b^{6} c c^{6} a c^{6} b} = 6 \sqrt[6]{a^{14} b^{14} c^{14}} = 6(abc)^{\frac{7}{3}} \\
\geqslant & 6(\sqrt{3})^{7} = 6 \times 27 \sqrt{3} = 162 \sqrt{3},
\end{aligned}$$

with equality holding if and only if $a = b = c = \sqrt{3}$. Thus, the minimum value is $162 \sqrt{3}$.