An invertible quotient of an invertible subsheaf by a torsion sheaf is a twist by an effective divisor

Local structure at a closed point. Fix a closed point $p$ and write $V={\mathcal{O}}_{C,p}$. Choose bases $e_L$ of ${\mathcal{L}}_p$ and $e_M$ of ${\mathcal{M}}_p$. The injection sends $e_L$ to $a{e}_M$ for a nonzero $a\in V$; writing $a=u{t}_p^k$ with $u\in V^{\times }$ by [F3], its image is $t_p^{k}V{e}_M$. Thus ${\mathcal{M}}_p\mathrm{/}{\mathcal{L}}_p\cong V\mathrm{/}(t_p^k)$, whose length is $k$ by [F3]. Since this quotient is ${\mathcal{Q}}_p$, $k=l_p$.

The generic point. Localizing the exact sequence at the generic point $\eta$ of the integral curve $C$ gives an exact sequence whose last term is the hypothesis-zero stalk ${\mathcal{Q}}_{\eta }=0$, so the morphism $\mathcal{L}\to \mathcal{M}$ restricts to an isomorphism ${\mathcal{L}}_{\eta }\to {\mathcal{M}}_{\eta }$ at the generic point. By [F2] both stalks are one-dimensional over $k(C)$, so this isomorphism is a nonzero rational trivialisation of the invertible sheaf $\mathcal{N}={\mathcal{L}}^{\vee }{\otimes }_{{\mathcal{O}}_C}\mathcal{M}$: the inclusion $\mathcal{L}\hookrightarrow \mathcal{M}$ is a nonzero morphism $\mathcal{L}\to \mathcal{M}$, hence a nonzero rational section $s$ of $\mathcal{N}$ in the sense of [F4].

The global divisor isomorphism. Let $D={\mathrm{div}}_C(s)$ and use [F5] to obtain the global isomorphism $\psi :{\mathcal{O}}_C(D)\to \mathcal{N}={\mathcal{L}}^{\vee }\otimes \mathcal{M}$ carrying $1_D$ to $s$. Tensoring by $\mathcal{L}$ and composing with the evaluation isomorphism $\mathcal{L}\otimes {\mathcal{L}}^{\vee }\cong {\mathcal{O}}_C$ gives a global isomorphism $\mathrm{\Phi }:\mathcal{L}\otimes {\mathcal{O}}_C(D)\to \mathcal{M}$. By the definition of $s$ from the original injection, the square comparing $\mathcal{L}\to \mathcal{M}$ with $\mathcal{L}\to \mathcal{L}\otimes {\mathcal{O}}_C(D)$, $\mathrm{\ell }\mapsto \mathrm{\ell }\otimes 1_D$, commutes: both maps send the generic section $\mathrm{\ell }$ to the original image of $\mathrm{\ell }$, and equality of maps to the locally free sheaf $\mathcal{M}$ can be checked at the generic point. Thus the isomorphism identifies the given subsheaf $\mathcal{L}$ with the canonical copy $\mathcal{L}\otimes {\mathcal{O}}_C\subseteq \mathcal{L}\otimes {\mathcal{O}}_C(D)$. On a trivializing open, the local equation of $D$ is the coefficient of the original regular morphism $\mathcal{L}\to \mathcal{M}$; its order at $p$ is $l_p$ by step 1.1. Thus the finite closed-point divisor notation for these local coefficients is $D={\sum }_p{l}_p[p]$ by [F6]. Its local equations are regular and nonzero, so it is effective by [F6]. Hence $\mathcal{M}\cong \mathcal{L}\otimes {\mathcal{O}}_C(D)$ as claimed.

The quotient sheaf and the finite-support trivialization. Put $N={\mathcal{O}}_C(D)\mathrm{/}{\mathcal{O}}_C$. Since step 2.1 identifies the inclusion $\mathcal{L}\to \mathcal{M}$ with $\mathcal{L}\to \mathcal{L}\otimes {\mathcal{O}}_C(D)$, taking cokernels gives the global isomorphism $\mathcal{Q}\cong \mathcal{L}\otimes N$. Its support is the finite set $S=\{p:l_p>0\}$, because $N_p\cong {\mathcal{O}}_{C,p}\mathrm{/}(t_p^{l_p})$ by the local divisor equation and [F3]. For each $p\in S$, choose an open neighborhood $U_p$ on which $\mathcal{L}$ is trivial and which contains no point of $S\setminus \{p\}$; such a neighborhood is obtained by intersecting a trivializing open with the complements of the finitely many other closed points of $S$. Also let $U_0=C\setminus S$. These opens cover $C$. On each $U_p$, a chosen frame of $\mathcal{L}$ gives an isomorphism $(\mathcal{L}\otimes N){\mathrm{\mid }}_{U_p}\cong N{\mathrm{\mid }}_{U_p}$, and on $U_0$ both sheaves vanish. For distinct $p,q\in S$, the intersection $U_p\cap U_q$ misses all of $S$, and $U_0\cap U_p$ also misses $S$, so both $N$ and $\mathcal{L}\otimes N$ vanish on every overlap between distinct members of this cover. The local isomorphisms therefore agree on overlaps and glue to a global (generally noncanonical) isomorphism $\mathcal{Q}\cong N={\mathcal{O}}_C(D)\mathrm{/}{\mathcal{O}}_C$.

Conclusion. For the effective divisor $D={\sum }_p{l}_p[p]$, the original inclusion and the rational-section isomorphism give $\mathcal{M}\cong \mathcal{L}\otimes {\mathcal{O}}_C(D)$, and the finite-support open-cover argument gives the noncanonical global isomorphism $\mathcal{M}\mathrm{/}\mathcal{L}\cong {\mathcal{O}}_C(D)\mathrm{/}{\mathcal{O}}_C$. The local lengths determine the coefficients, while the latter isomorphism also uses the finite support and chosen trivializations of $\mathcal{L}$ near that support.