This is part of Proposition 3.6.2 in Mazorchuk's Lectures on $\mathfrak{sl}_2$-modules.
He denotes by $\overline{\mathfrak{W}}^{\xi,\tau}$ the category of weight $\mathfrak{sl}_2$-modules with support in $\xi\in\mathbb{C}/2\mathbb{Z}$, such that $M$ is spanned by weight vectors which are generalized eigenvectors of the Casimir element $c$, with eigenvalue $\tau$. He's proving that if $M\in\overline{\mathfrak{W}}^{\xi,\tau}$, then $M$ has finite length. He takes $L\in\overline{\mathfrak{W}}^{\xi,\tau}$ a simple module, and it's known the non-zero weight spaces of $L$ are one dimensional by a classification theorem. Then he says the composition multiplicity $[M:L]$ of $L$ in $M$ cannot exceed $\dim M_\lambda$ for any $\lambda\in\mathbb{C}$ such that $\dim L_\lambda\neq 0$. Here $M_\lambda$ is the weight space of $M$ with weight $\lambda$, etc. Why is this restriction true? I can understand his reasoning after this point.
As I explain in my answer to your other question, we have $$\mathrm{ch}M=\sum_L[M:L]\mathrm{ch}L$$ where $L$ runs over all simple $\mathfrak{g}$-modules. In particular, $$\dim M_\lambda=\sum_L[M:L]\dim L_\lambda.$$ As $\dim L_\lambda\in\{0,1\}$, $$\dim M_\lambda=\sum_{L:L_\lambda\neq0}[M:L].$$ Therefore, if $S$ is any simple module with $S_\lambda\neq 0$, we must have $$\dim M_\lambda\geq [M:S]$$