  Coexisting density-wave and superconducting states along with the large
anomalous Hall effect in the absence of local magnetism remain intriguing and
enigmatic features of the AV$_3$Sb$_5$ kagome metals (A = K, Rb, Cs). Here, we
demonstrate via optical spectroscopy and density-functional calculations that
low-energy dynamics of KV$_3$Sb$_5$ is characterized by unconventional
localized carriers, which are strongly renormalized across the density-wave
transition and indicative of electronic correlations. Strong phonon anomalies
are prominent not only below the density-wave transition, but also at high
temperatures, suggesting an intricate interplay of phonons with the underlying
electronic structure. We further propose the star-of-David and tri-hexagon
(inverse star-of-David) configurations for the density-wave order in
KV$_3$Sb$_5$. These configurations are strongly reminiscent of $p$-wave states
expected in the Hubbard model on the kagome lattice at the filling level of the
van Hove singularity. The proximity to this regime should have intriguing and
far-reaching implications for the physics of KV$_3$Sb$_5$ and related
materials.
