The Axiom of Choice and the Cartesian Product.

Suppose $X=\{X_i\mid i\in I\}$ is a family of nonempty sets.

If there exists a choice function, then $\langle f(i)\mid i\in I\rangle$ is an element of the product $\prod_{i\in I}X_i$.

If $\prod_{i\in I}X_i$ is nonempty then there is $f=\langle x_i\mid i\in I\rangle$ in this product, which is a sequence of $x_i$ such that $x_i\in X_i$. The function $f(i)=x_i$ is a choice function.

Indeed as Nate comments, it is most common to define the product $\prod_{i\in I}X_i$ as the set of functions $f:I\to\bigcup\{X_i\mid i\in I\}$ such that $f(i)\in X_i$ for all $i\in I$.

One can easily observe that under this definition the product is exactly the set of choice functions, therefore the product is nonempty if and only if there exists a choice function.