A language $L$ describes decision problem $p$ means that $\forall x \in L $ the instance $x$ is a Yes/Accept instance of $p$.
For the independent set problem, we have to define language $L$ such that every $x\in L$ is an independent set of graph $G=(V, E)$.
So one possible way to define decision language for independent set problem is
$$\begin{array}{ll}
L & = \{I \mbox{ such that } I\subseteq V \mbox{ and }\forall u,v\in I,\;\; (u,v)\notin E\} \\
& = \{I \;\;|\;\; I\subseteq V \mbox{ and }\forall u,v\in I, \;\; (u,v)\notin E\}
\end{array}
.$$
If you wish to define the language generally you can write it as
$$ \begin{array}{ll}
L & =\{(G=(V,E),I) \mbox{ such that } I\subseteq V \mbox{ and }\forall u,v\in I,\;\; (u,v)\notin E\}\\
& =\{(G=(V,E),I) \;\;|\;\; I\subseteq V \mbox{ and }\forall u,v\in I,\;\; (u,v)\notin E\}
\end{array}
.$$