Example 2 Find all increasing functions $f: \mathbf{N} \rightarrow \mathbf{Z}$ satisfying:
$$
\begin{array}{l}
f(2)=7, \\
f(m n)=f(m)+f(n)+f(m) f(n) .
\end{array}
$$
(2014, Taiwan Mathematical Olympiad Training Camp)
Solving: In equation (1), taking $m=0, n=2$, we get
$$
f(2)+f(0) f(2)=0 \text{.}
$$

Since $f(2)=7$, it follows that $f(0)=-1$.
In equation (1), taking $m=1, n=2$, we get
$$
f(1)+f(1) f(2)=0 \text{.}
$$

Since $f(2)=7$, it follows that $f(1)=0$.
For $n \geqslant 2$, let $g(n)=f(n)+1$.
Thus, $g(2)=8$,
$$
\begin{array}{l}
g(m n)=f(m n)+1 \\
=(f(m)+1)(f(n)+1) \\
=g(m) g(n) .
\end{array}
$$

By the variant of Cauchy's functional equation (3), we know
$$
g(n)=n^{a} \text{.}
$$

Since $g(2)=8$, then $a=3$, i.e., $f(n)=n^{3}-1$. Upon verification, $f(n)=n^{3}-1$ is a solution to the original equation.