  Among transition metal dichalcogenides (TMDs), VSe$_2$ is considered to
develop a purely 3-dimensional (3D) charge-density wave (CDW) at T$_{CDW}$=110
K. Here, by means of high resolution inelastic x-ray scattering (IXS), we show
that the CDW transition is driven by the collapse of an acoustic mode at the
critical wavevector \textit{q}$_{CDW}$= (2.25 0 0.7) r.l.u. and critical
temperature T$_{CDW}$=110 K. The softening of this mode starts to be pronounced
for temperatures below 2$\times$ T$_{CDW}$ and expands over a rather wide
region of the Brillouin zone, suggesting a large contribution of the
electron-phonon interaction to the CDW formation. This interpretation is
supported by our first principles calculations that determine a large
momentum-dependence of the electron-phonon interaction, peaking at the CDW
wavevector, in the presence of nesting. Fully anharmonic {\it ab initio}
calculations confirm the softening of one acoustic branch at \textit{q}$_{CDW}$
as responsible for the CDW formation and show that van der Waals interactions
are crucial to melt the CDW. Our work also highlights the important role of
out-of-plane interactions to describe 3D CDWs in TMDs.
