  The coexistence of the charge-density wave (CDW) and superconducting phases
and their tunability under external pressure remains one of the key points in
understanding the electronic structure of $A$V$_3$Sb$_5$ ($A$ = K, Rb, Cs)
kagome metals. Here, we employ synchrotron-based infrared spectroscopy assisted
by density-functional calculations to study the pressure evolution of the
electronic structure at room temperature up to 17 GPa experimentally. The
optical spectrum of CsV$_3$Sb$_5$ is characterized by the presence of localized
carriers seen as a broad peak at finite frequencies in addition to the
conventional metallic Drude response. The pressure dependence of this
low-energy peak reflects the re-entrant behavior of superconductivity and may
be interpreted in terms of electron-phonon coupling, varying with the growth
and shrinkage of the Fermi surface. Moreover, drastic modifications in the
low-energy interband absorptions are observed upon the suppression of CDW.
These changes are related to the upward shift of the Sb2 $p_x+p_y$ band that
eliminates part of the Fermi surface around the $M$-point, whereas band saddle
points do not move significantly. These observations shed new light on the
mixed electronic and lattice origin of the CDW in CsV$_3$Sb$_5$.
