If $L\in\mathrm{NP}$ is $\mathrm{coNP}$-complete, then $\mathrm{coNP}=\mathrm{NP}$
$\newcommand{\co}{\mathsf{co}\text{-}}$

Let $A\subseteq \Sigma^*$ be some class of languages. Saying $\co{A}\subseteq A$ means that $A$ is closed under complement, and this is equivalent to $\co{A}=A$.

To show the other direction, $A\subseteq \co{A}$, let $L\in A$ be some language in the class $A$. $\overline{L}\in \co{A}$, and since we know $\co{A}\subseteq A$, we have $\overline{L}\in A$, which means $L\in \co{A}$.