  The recent discovery of superconductivity in La$_3$Ni$_2$O$_7$ with
$T_\mathrm{c} \simeq 80~\mathrm{K}$ under high pressure opens up a new route to
high-$T_\mathrm{c}$ superconductivity. This material realizes a bilayer square
lattice model featuring a strong interlayer hybridization unlike many
unconventional superconductors. A key question in this regard concerns how
electronic correlations driven by the interlayer hybridization affect the
low-energy electronic structure and the concomitant superconductivity. Here, we
demonstrate using a cluster dynamical mean-field theory that the interlayer
electronic correlations (IECs) induce a Lifshitz transition resulting in a
change of Fermi surface topology. By solving an appropriate gap equation, we
further show that the leading pairing instability, $s \pm$-wave, is enhanced by
the IECs. The underlying mechanism is the quenching of a strong ferromagnetic
channel, resulting from the Lifshitz transition driven by the IECs. Based on
this picture, we provide a possible reason of why superconductivity emerges
only under high pressure.
