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The low-temperature state of the quantum spin liquid candidate κ-(BEDT-TTF)_{2}Cu_{2}(CN)_{3} emerges via an anomaly at T^{*}∼6 K. Although signatures of this anomaly have been revealed in various quantities, its origin has remained unclear. Here we report inelastic neutron scattering measurements on single crystals of κ-(BEDT-TTF)_{2}Cu_{2}(CN)_{3}, aiming at studying phonon renormalization effects at T^{*}. A drastic change was observed in the phonon damping across T^{*} for a breathing mode of BEDT-TTF dimers at E=4.7 meV. The abrupt change in the phonon damping is attributed to a phase transition into a valence bond solid state based on an effective model describing the spin-charge coupling in this dimer-Mott system.