The sentence "$f$ increases most in the direction $Df(x_0)$" is misleading and hide numerous technical details. Formally, $Df(x_0)$ is a linear operator, the differential of $f$ at a point, not a vector. The Jacobian matrix of $f \in C^1(R^n, R^m)$ is the linear map representing the differential. If $n=m=1$, the differential is simply what we call the derivative of $f$ and if $m=1$, the Jacobian matrix contains a unique line vector. In the end, in the specific case $f ∈ C^1(R^n, R)$, the direction of maximal increase is given by the transpose of the Jacobian Matrix, $Df(x_0)^T$, or represen. Indeed, the Jacobian matrix is made of partial derivatives, and as $f \in C^1(R^n, R)$, directional derivatives are deduced from the partial derivatives.
So, as explained, $||Df(x_0)||$ should not be understood as a norm of a vector but as a norm of a linear operator, or a matrix in this case. Since it is still a norm in the usual sense, $||Df(x_0)||=0$ implies that this linear operator is the zero operator, i.e. all partial derivatives are $0$, hence all directional derivatives as well. Conversely, if $||Df(x_0)|| \ne 0$, then there is at least one partial derivative that is non-zero. There are many norms on matrices depending on the context, some of them are described here.