  The Fermi Gamma-ray Space Telescope recently detected the most energetic
gamma-ray burst so far, GRB 080916C, and reported its detailed temporal
properties in an extremely broad spectral range: (i) the time-resolved spectra
are well described by broken power-law forms over the energy range of $10 {\rm
keV}-10$ GeV, (ii) the high-energy emission (at $\epsilon > 100$ MeV) is
delayed by $\approx 5$s with respect to the $\epsilon \lesssim 1$ MeV emission,
and (iii) the emission onset times shift towards later times in the higher
energy bands. We show that this behavior of the high-energy emission can be
explained by a model in which the prompt emission consists of two components:
one is the emission component peaking at $\epsilon \sim 1$ MeV due to the
synchrotron-self-Compton radiation of electrons accelerated in the internal
shock of the jet and the other is the component peaking at $\epsilon \sim 100$
MeV due to up-scattering of the photospheric X-ray emission of the expanding
cocoon (i.e., the hot bubble produced by dissipation of the jet energy inside
the progenitor star) off the same electrons in the jet. Based on this model, we
derive some constraints on the radius of the progenitor star and the total
energy and mass of the cocoon of this GRB, which may provide information on the
structure of the progenitor star and the physical conditions of the jet
propagating in the star. The up-scattered cocoon emission could be important
for other Fermi-GRBs as well. We discuss some predictions of this model,
including a prompt bright optical emission and a soft X-ray excess.
