  We develop a microscopic theory of dynamic structure factor to describe the
Bogoliubov-Anderson-Goldstone phonon mode and its damping rate in a strongly
interacting Fermi gas at finite temperature. It is based on a density
functional approach - the so-called superfluid local density approximation. The
accuracy of the theory is quantitatively examined by comparing the theoretical
predictions with the recent experimental measurements for the local dynamic
structure factor of a nearly homogeneous unitary Fermi gas at low transferred
momentum {[}S. Hoinka \textit{et al.}, Nat. Phys. \textbf{13}, 943 (2017){]},
without any free parameters. We calculate the dynamic structure factor as
functions of temperature and transferred momentum, and determine the
temperature evolution of the phonon damping rate, by considering the dominant
decay process of the phonon mode via scatterings off fermionic quasiparticles.
These predictions can be confronted with future Bragg scattering experiments on
a unitary Fermi gas near the superfluid transition.
