  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
$T^\ast\simeq 155$ 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$_2$.
