Let $D_d$ be the little $d$-disk operad as outlined in Fresse's book Homotopy of Operads and Grothendieck-Teichmuller Groups.
We have the sequence of inclusions of operads $$D_1 \to D_2 \to \cdots \to D_n \to \cdots$$
The operad $D_\infty$ was set as $colim_n D_n$. I'm not sure about the notation $colim_n D_n$. I'm assuming that it's the colimit of the diagram $$D_1 \to D_2 \to \cdots \to D_n \to \cdots$$
Is it correct?
Yes, it's the colimit of this diagram. Concretely, $D_\infty(k)$ is a quotient of the disjoint union $\bigsqcup_{n \ge 1} D_n(k)$ under some equivalence relation. The equivalence relation is generated by the identification of $x \in D_n(k)$ with its image in $D_{n+1}(k)$ under the inclusion map. Then you can check that the operadic structure maps are compatible with this equivalence relation (essentially because the inclusions $D_n \to D_{n+1}$ are operad morphisms) and thus you get an operad structure on $D_\infty$.