Let $k$ be a field and $\mathscr{F}$ a coherent sheaf on $\mathbf{P}^n_k$. In paragraph $5.2$ of Fundamental Algebraic Geometry, it is claimed that if $H\subseteq\mathbf{P}^n_k$ is a hyperplane which does not contain any associated point of $\mathscr{F}$, then one has a short exact sequence:
$0\rightarrow \mathscr{F}(-1)\rightarrow\mathscr{F}\rightarrow\mathscr{F}_H\rightarrow 0$
where $\mathscr{F}_H$ is the restriction of $\mathscr{F}$ to $H$ (I guess the author rather means $i_*\mathscr{F}_H$ where $i$ is the closed immersion $H\rightarrow\mathbf{P}^n_k$) and the first arrow $\mathscr{F}(-1)\rightarrow \mathscr{F}$ is locally given by multiplication with a defining equation of $H$.
Why is this true? Actually, I have no idea on how the assumption on associated points of $\mathscr{F}$ is used here.