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 $zsim2$, 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 $zsim0.6$ versus $zsim2$. 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.
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