Let $\mathbf{x}\in \mathbb{R}^n$, and let $\|\mathbf{x}\|_L$ be the sum of the $L$ largest absolute components of $\mathbf{x}$. That is to say, write the (absolute values of) components of $\mathbf{x}$ in descending order and sum the first $L$ components. This gives you the norm in question.
Now how is this norm equivalent to the following linear program? \begin{equation*} \begin{aligned} & \underset{\boldsymbol{\eta}}{\text{maximize}} && \Sigma_{i=1}^{n}|x_i|\,\eta_i \\ & \text{subject to} && \boldsymbol{\eta}^T\mathbf{1}=L\\ &&& \mathbf{0}\leq\boldsymbol{\eta}\leq \mathbf{1} \end{aligned} \end{equation*}
I guess it should be sufficient to show that the objective function with the given constraints is equivalent to the given norm?
(Courtesy: Comment by Michal Adamaszek)
First, ignore the equality constraint and assume that the $\boldsymbol{\eta}$ is subject to inequality constraints only i.e. its components can only take on values between $0$ and $1$. So, how to maximize the objective function which is affine in $\boldsymbol{\eta}$ and has non-negative weights $|x_i|$? Clearly, $\boldsymbol{\eta}=\mathbf{1}$ will be optimal in this case. Now, also enforce the equality constraint i.e. at most $L$ of the components of $\boldsymbol{\eta}$ can take on the value $1$. Now which components would these be? To maximize the objective, these will precisely be the ones corresponding to the $L$ largest components of $|\mathbf{x}|$. Thus, the optimal value returned by this LP is same as the norm $||\mathbf{x}||_L$.