Here I'm asumming $\partial E = \{ x : \text{ every open ball around $x$ contains points of $E$ and $E^c$}\}$
Suppose $\partial E \subseteq E$. Then let $x\in E^c$, then since $\partial E\subset E$ we must have some open ball which contains only points of $E^c$ around $x$, so $E^c$ is open, and hence $E$ is closed.
Now suppose that $E$ is closed. Then $E^c$ is open, so for every $x\in E^c$ we have an open ball around $x$ which is contained completely in $E^c$. This means that $E^c \cap \partial E = \emptyset$, and hence $\partial E \subset E$.