  We report optical measurements demonstrating that the low-energy relaxation
rate ($1/\tau$) of the conduction electrons in Sr$_2$RuO$_4$ obeys scaling
relations for its frequency ($\omega$) and temperature ($T$) dependence in
accordance with Fermi-liquid theory. In the thermal relaxation regime,
$1/\tau\propto (\hbar\omega)^2 + (p\pi\kB T)^2$ with $p=2$, and $\omega/T$
scaling applies. Many-body electronic structure calculations using dynamical
mean-field theory confirm the low-energy Fermi-liquid scaling, and provide
quantitative understanding of the deviations from Fermi-liquid behavior at
higher energy and temperature. The excess optical spectral weight in this
regime provides evidence for strongly dispersing "resilient" quasiparticle
excitations above the Fermi energy.
