EuAl4 is a rare-earth intermetallic in which competing itinerant and/or indirect exchange mechanisms give rise to a complex magnetic phase diagram, including a centrosymmetric skyrmion lattice. These phenomena arise not in the tetragonal parent structure but in the presence of a charge-density wave (CDW), which lowers the crystal symmetry and renormalizes the electronic structure. Microscopic knowledge of the corresponding atomic modulations and their driving mechanism is a prerequisite for a deeper understanding of the resulting equilibrium of electronic correlations and how it might be manipulated. Here, we use synchrotron single-crystal x-ray diffraction, inelastic x-ray scattering, and lattice-dynamics calculations to clarify the origin of the CDW in EuAl4. We observe a broad softening of a transverse acoustic phonon mode that sets in well above room temperature and, at TCDW=142 K, freezes out in an atomic displacement mode described by the superspace group Immm(00γ)s00. In the context of previous work, our observation is a clear confirmation that the CDW in EuAl4 is driven by electron-phonon coupling. This result is relevant for a wider family of BaAl4 and ThCr2Si2-type rare-earth intermetallics known to combine CDW instabilities and complex magnetism.