What is the simplest way to understand $\nabla \cdot (\nabla \times \textbf{A} ) =0$? Of course one can prove it by brute force.
But how to Understand it, in the simplest way?
Another issue is $\nabla \times (\nabla f) =0 $.

It's just a way to remember/understand. Think that $\nabla $ is a vector. Then, by definition, $\nabla \times A$ is perpendicular to $\nabla $ and so the product scalar of $\nabla $ and $\nabla \times A$ is nul, therefore
$$\nabla \cdot  (\nabla \times A)=0$$
