  We present spatially resolved Hubble Space Telescope grism spectroscopy of 15
galaxies at $z\sim0.8$ drawn from the DEEP2 survey. We analyze H$\alpha$+[N
II], [S II] and [S III] emission on kpc scales to explore which mechanisms are
powering emission lines at high redshifts, testing which processes may be
responsible for the well-known offset of high redshift galaxies from the
$z\sim0$ locus in the [O III]/H$\beta$ versus [N II]/H$\alpha$ BPT
(Baldwin-Phillips-Terlevich) excitation diagram. We study spatially resolved
emission line maps to examine evidence for active galactic nuclei (AGN),
shocks, diffuse ionized gas (DIG), or escaping ionizing radiation, all of which
may contribute to the BPT offsets observed in our sample. We do not find
significant evidence of AGN in our sample and quantify that, on average, AGN
would need to contribute $\sim$25% of the H$\alpha$ flux in the central
resolution element in order to cause the observed BPT offsets. We find weak
($2\sigma$) evidence of DIG emission at low surface brightnesses, yielding an
implied total DIG emission fraction of $\sim$20%, which is not significant
enough to be the dominant emission line driver in our sample. In general we
find that the observed emission is dominated by star forming H II regions. We
discuss trends with demographic properties and the possible role of
$\alpha$-enhanced abundance patterns in the emission spectra of high redshift
galaxies. Our results indicate that photo-ionization modeling with stellar
population synthesis inputs is a valid tool to explore the specific star
formation properties which may cause BPT offsets, to be explored in future
work.
