The optical properties of KFe$_2$As$_2$ have been measured for light polarized in the a-b planes over a wide temperature and frequency range. Below $T^astsimeq 155$ K, where this material undergoes an incoherent-coherent crossover, we observe a new coherent response emerging in the optical conductivity. A spectral weight analysis suggests that this new feature arises out of high-energy bound states. Below about $T_{rm FL} simeq 75$ K the scattering rate for this new feature is quadratic in temperature, indicating a Fermi-liquid response. Theory calculations suggest this crossover is dominated by the $d_{xy}$ orbital. Our results advocate for Kondo-type screening as the mechanism for the orbital-selective incoherent-coherent crossover in hole-overdoped KFe$_2$As$_2$.
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