Say your function is $T\colon V\to W$. Then the codomain of this function is $W$ and the range of $T$ is the subset of all $w\in W$ which are actually realized by the function $T$.
That is, $w\in\textrm{range}(T)$ if and only if there exists an element $v\in V$ with $T(v)=w$. Note that $W=\textrm{range}(T)$ is not necessarily true, it might be that $\textrm{range}(T)\subsetneq W$. Therefore the elements in $\textrm{range}(T)$ need to be characterized by a special property.
So in the notation $\left\\{\cdot | \cdot\right\\}$ or $\left\\{\cdot : \cdot\right\\}$, the part after the "$|$" or the "$:$" simply describes the property that an element of $W$ needs to satisfy in order to be contained in this set. Whether you read this as _such that_ or _for which_ or _with the property that_ is your personal decision and a matter of taste.