Our polarization-resolved Raman spectra of the kagome metal CsV₃Sb₅ show a small but significant anisotropy of a large gap in the electronic excitation spectrum below the charge density wave (CDW) transition at T_CDW = 95K which escaped detection so far. The projections observed in A_1g and E_2g symmetry suggest that a gap may open up also on the Fermi surface encircling the Γ-point. The simulations using density-functional theory (DFT) reproduce the observed spectral changes and show that both energy considerations and spectroscopy support the tri-hexagonal rather than the Star of David distortion which cannot be distinguished on the basis of the phonon selection rules. The temperature-dependent line width of the low-energy optical A_1g mode indicates enhanced phonon-phonon coupling below T_CDW. The A_1g amplitude mode (AM) develops an unexpected Fano- type line shape which describes the strong coupling of an isolated oscillator and a continuum. The large electronic gap, the enhanced anharmonic phonon-phonon coupling, and the Fano shape of the AM combined are more supportive of a strong-coupling phonon-driven CDW transition than of a Fermi surface instability or an exotic mechanism.