It is the classifying category for the left action of $C$ on its product $C \times C$.

Let me elaborate on this a bit further. Let $C$ be an $E_\infty$-monoid, for example represented by a symmetric monoidal groupoid (which is the case for Quillen's Applications of e.g. $Proj(R)$). We can then consider the symmetric monoidal $\infty$-category $B C$ given by a single object and mapping space equal to $C$.There's a natural functor $C^\circlearrowleft : B C \rightarrow Spc$, essentially given by the functor corepresented by the single object. Here $Spc$ means the $\infty$-category of spaces.

Then we can consider the functor $C^\circlearrowleft \times C^\circlearrowleft : B C \rightarrow Spc$, and take it's unstraightening, i.e. Grothendieck construction. This is given by the functor $C^{-1} C \rightarrow B C$ which sends the arrow $s+ : (n,m) \rightarrow (s+n,s+m)$ in $C^{-1} C$ to the arrow $* \xrightarrow{s} *$ in $B C$.

An almost trivial consequence of this is that the realization of $C^{-1} C$ is just the ($\infty$-categorical) colimit $(C^\circlearrowleft \times C^\circlearrowleft)_{hC}$.

This construction exists for an arbitrary $E_n$-monoid, but needs Quillen's additional assumptions that we start with a symmetric monoidal groupoid on which addition by an object is faithful to make sure that $C^{-1} C$ is actually (equivalent to) a 1-category.

It is the classifying category for the left action of $C$ on its product $C \times C$.

Let me elaborate on this a bit further. Let $C$ be an $E_\infty$-monoid, for example represented by a symmetric monoidal groupoid (which is the case for Quillen's applications of e.g. $Proj(R)$). We can then consider the symmetric monoidal $\infty$-category $B C$ given by a single object and mapping space equal to $C$.There's a natural functor $C^\circlearrowleft : B C \rightarrow Spc$, essentially given by the functor corepresented by the single object. Here $Spc$ means the $\infty$-category of spaces.

Then we can consider the functor $C^\circlearrowleft \times C^\circlearrowleft : B C \rightarrow Spc$, and take its unstraightening, i.e. Grothendieck construction. This is given by the functor $C^{-1} C \rightarrow B C$ which sends the arrow $s+ : (n,m) \rightarrow (s+n,s+m)$ in $C^{-1} C$ to the arrow $* \xrightarrow{s} *$ in $B C$.

An almost trivial consequence of this is that the realization of $C^{-1} C$ is just the ($\infty$-categorical) colimit $(C^\circlearrowleft \times C^\circlearrowleft)_{hC}$.

This construction exists for an arbitrary $E_n$-monoid, but needs Quillen's additional assumptions that we started with a symmetric monoidal groupoid on which addition by an object is faithful to make sure that $C^{-1} C$ is actually (equivalent to) a 1-category.