Example 1 Let non-negative real numbers $a_{1}, a_{2}, \cdots, a_{n}$ satisfy $a_{1}+$ $a_{2}+\cdots+a_{n}=1$, find
$$
\begin{array}{l} 
\frac{a_{1}}{1+a_{2}+\cdots+a_{n}}+\frac{a_{2}}{1+a_{1}+a_{3}+\cdots+a_{n}}+\cdots \\
+\frac{a_{n}}{1+a_{1}+\cdots+a_{n-1}} \text { 's minimum value. }
\end{array}
$$