Optical conductivity measurements are combined with density functional theory calculations in order to understand the electrodynamic response of the frustrated Mott insulators Herbertsmithite $mathrm{ZnCu_{3}(OH)_{6}Cl_{2}}$ and the closely-related kagome-lattice compound $mathrm{Y_{3}Cu_{9}(OH)_{19}Cl_{8}}$. We identify these materials as charge-transfer rather than Mott-Hubbard insulators, similar to the high-$T_c$ cuprate parent compounds. The band edge is at 3.3 and 3.6 eV, respectively, establishing the insulating nature of these compounds. Inside the gap, we observe dipole-forbidden local electronic transitions between the Cu $3d$ orbitals in the range 1--2 eV. With the help of textit{ab initio} calculations we demonstrate that the electrodynamic response in these systems is directly related to the role of on-site Coulomb repulsion: while charge-transfer processes have their origin on transitions between the ligand band and the Cu $3d$ upper Hubbard band, textit{local} $d$-$d$ excitations remain rather unaffected by correlations.
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