Following Martin's suggestion, I will turn my comment into an answer.

If $T$ is an Grothendieck topos, then the subobjects of the terminal object form a frame. If $X_T$ is the corresponding locale, then the topos $Sh(X_T)$ of sheaves on $X_T$ is the called localic reflection of $T$. One has that $T\mapsto X_T$ is adjoint to the functor that takes a locale to its topos of sheaves. Moreover, if $X$ is any locale then $Sh(X)$ is equivalent to its localic reflection. If $X$ is a topological space, then $Sh(X)$ is equivalent to the category of sheaves on the locale corresponding to its frame $O(X)$ of open subsets. It is in this sense that you can recover the locale of open subsets of $X$.

Now associated to any locale $L$ is its space of points $Pts(L)$. Let $O(L)$ denote the frame corresponding to $L$. The functor $L$ to $Pts(L)$ is adjoint to the functor taking a space to its corresponding locale. A point of $L$ is a homomorphism of frames $p\colon O(L)\to O(\Sigma)$ where $\Sigma$ is the Sierpinski space $\{0,1\}$ with open subsets $\emptyset, \{1\}, \{0,1\}$. A locale may not have any points. The topology on $O(L)$ is the usual "Zariski" topology: a basic open set is of the form $D(U)$ with $U\in O(L)$ where $D(U)$ consists of those points $p$ with $p(U)=1$.

If $X$ is a space, then to each point $x\in X$, we get a point $p_x$ given by

$p_x(U)=\begin{cases} 1 & x\in U\\ 0 & x\notin U\end{cases}$.

Notice that these points suffice to separate open sets (i.e., the locale of a space has enough points). The map $x\to p_x$ is injective iff $X$ is $T_0$ and is an isomorphism iff $X$ is a sober space (each irreducible closed subset has a unique generic point). Let $Sob(X)$ be the space of points of the locale of the space $X$. Note that $O(X)$ and $O(Sob(X))$ are isomorphic frames. The natural map $X\to Sob(X)$ is continuous and is the universal map of $X$ into a sober space. Sometimes $Sob(X)$ is called the soberification of $X$. Note that $X$ is sober iff the natural map is a homeomorphism. Therefore, the information that can be recovered about a space $X$ from its topos of sheaves is the space $Sob(X)$ (up to homeomorphism).

Following Martin's suggestion, I will turn my comment into an answer.

If $T$ is an Grothendieck topos, then the subobjects of the terminal object form a frame. If $X_T$ is the corresponding locale, then the topos $Sh(X_T)$ of sheaves on $X_T$ is the called localic reflection of $T$. One has that $T\mapsto X_T$ is adjoint to the functor that takes a locale to its topos of sheaves. Moreover, if $X$ is any locale then $Sh(X)$ is equivalent to its localic reflection. If $X$ is a topological space, then $Sh(X)$ is equivalent to the category of sheaves on the locale corresponding to its frame $O(X)$ of open subsets. It is in this sense that you can recover the locale of open subsets of $X$.

Now associated to any locale $L$ is its space of points $Pts(L)$. Let $O(L)$ denote the frame corresponding to $L$. The functor $L$ to $Pts(L)$ is adjoint to the functor taking a space to its corresponding locale. A point of $L$ is a homomorphism of frames $p\colon O(L)\to O(pt)$ where $pt$ is the one point space. It is convenient to call the two elements of $O(pt)$ $0,1$ with $0<1$. A locale may not have any points. The topology on $O(L)$ is the usual "Zariski" topology: a basic open set is of the form $D(U)$ with $U\in O(L)$ where $D(U)$ consists of those points $p$ with $p(U)=1$.

If $X$ is a space, then to each point $x\in X$, we get a point $p_x$ given by

$p_x(U)=\begin{cases} 1 & x\in U\\ 0 & x\notin U\end{cases}$.

Notice that these points suffice to separate open sets (i.e., the locale of a space has enough points). The map $x\to p_x$ is injective iff $X$ is $T_0$ and is an isomorphism iff $X$ is a sober space (each irreducible closed subset has a unique generic point). Let $Sob(X)$ be the space of points of the locale of the space $X$. Note that $O(X)$ and $O(Sob(X))$ are isomorphic frames. The natural map $X\to Sob(X)$ is continuous and is the universal map of $X$ into a sober space. Sometimes $Sob(X)$ is called the soberification of $X$. Note that $X$ is sober iff the natural map is a homeomorphism. Therefore, the information that can be recovered about a space $X$ from its topos of sheaves is the space $Sob(X)$ (up to homeomorphism).