  The nature of the superconducting pairing state in the pristine phase of a
compressed kagome metal CsV$_3$Sb$_5$ under pressure is studied by the
Migdal-Eliashberg formalism and density-functional theory calculations. We find
that the superconducting gap distribution driven by electron-phonon coupling is
nodeless and anisotropic. It is revealed that the hybridized V 3$d$ and Sb 5$p$
orbitals are strongly coupled to the V-V bond-stretching and V-Sb bond-bending
phonon modes, giving rise to a wide spread of superconducting gap depending on
its associated Fermi-surface sheets and momentum. Specifically, the
superconducting gaps associated with V 3$d_{xy,x^2-y^2,z^2}$ and 3$d_{xz,yz}$
orbitals are larger in their average magnitude and more widely spread compared
to that associated with the Sb 5$p_z$ orbital. Our findings demonstrate that
the superconductivity of compressed CsV$_3$Sb$_5$ can be explained by the
anisotropic multiband pairing mechanism with conventional phonon-mediated
$s$-wave symmetry, evidenced by recent experimental observations under pressure
as well as at ambient pressure.
