Solution:

By the inequality $a+b \geq 2 \sqrt{a b}$ which holds for positive numbers $a, b$ (and equality is if and only if $a=b$), we see that $1+a_{1} \geq 2 \sqrt{a_{1}}$, $1+a_{2} \geq 2 \sqrt{a_{2}}$, $\ldots$, $1+a_{n} \geq 2 \sqrt{a_{n}}$. Multiplying these inequalities we get
$$\left(1+a_{1}\right) \cdots \left(1+a_{n}\right) \geq 2^{n} \sqrt{a_{1} \cdots a_{n}} = 2^{n}.$$
The equality holds if and only if $a_{1}=a_{2}=\cdots=a_{n}=1$.