A classical $R$-matrix $r\in {\rm End}(\mathfrak{g}\otimes \mathfrak{g})$ acts on 2 copies of a Lie algebra $\mathfrak{g}$.
The Sweedler notation $r_{k\ell}\in {\rm End}(\mathfrak{g}\otimes \mathfrak{g}\otimes \mathfrak{g})$ means that $r_{k\ell}$ acts on the triple tensor product $\mathfrak{g}\otimes \mathfrak{g}\otimes \mathfrak{g}$ by letting $r$ acting on the $k$'th and $\ell$'th copy of the Lie algebra $\mathfrak{g}$, and an identity ${\bf 1}_{\mathfrak{g}}$ acts on the remaining copy.
The notation for the quantum $R$-matrix $R$ and the quantum Yang-Baxter equation is similar.