  We present multi-band Hubble Space Telescope imaging that spans rest-frame
near-ultraviolet through near-infrared wavelengths (0.3-1.1 $\mu$m) for 12
compact starburst galaxies at z=0.4-0.8. These massive galaxies (M_stellar ~
10^11 M_Sun) are driving very fast outflows ($v_{max}$=1000-3000 km/s), and
their light profiles are dominated by an extremely compact starburst component
(half-light radius ~ 100 pc). Our goal is to constrain the physical mechanisms
responsible for launching these fast outflows by measuring the physical
conditions within the central kiloparsec. Based on our stellar population
analysis, the central component typically contributes $\approx$25% of the total
stellar mass and the central escape velocities $v_{esc,central}\approx900$ km/s
are a factor of two smaller than the observed outflow velocities. This requires
physical mechanisms that can accelerate gas to speeds significantly beyond the
central escape velocities, and it makes clear that these fast outflows are
capable of traveling into the circumgalactic medium, and potentially beyond. We
find central stellar densities comparable to theoretical estimates of the
Eddington limit, and we estimate $\Sigma_1$ surface densities within the
central kpc comparable to those of compact massive galaxies at $0.5<z<3.0$.
Relative to "red nuggets" and "blue nuggets" at $z\sim2$, we find significantly
smaller $r_e$ values at a given stellar mass, which we attribute to the
dominance of a young stellar component in our sample and the better physical
resolution for rest-frame optical observations at $z\sim0.6$ versus $z\sim2$.
We compare to theoretical scenarios involving major mergers and violent disc
instability, and we speculate that our galaxies are progenitors of power-law
ellipticals in the local universe with prominent stellar cusps.
