  We analyze the colors and sizes of 32 quiescent (UVJ-selected) galaxies with
strong Balmer absorption ($\mbox{EW}(H\delta) \geq 4$\AA) at $z\sim0.8$ drawn
from DR2 of the LEGA-C survey to test the hypothesis that these galaxies
experienced compact, central starbursts before quenching. These recently
quenched galaxies, usually referred to as post-starburst galaxies, span a wide
range of colors and we find a clear correlation between color and half-light
radius, such that bluer galaxies are smaller. We build simple toy models to
explain this correlation: a normal star-forming disk plus a central, compact
starburst component. Bursts with exponential decay timescale of $\sim$~100 Myr
that produce $\sim10\%$ to more than 100\% of the pre-existing masses can
reproduce the observed correlation. More significant bursts also produce bluer
and smaller descendants. Our findings imply that when galaxies shut down star
formation rapidly, they generally had experienced compact, starburst events and
that the large, observed spread in sizes and colors mostly reflects a variety
of burst strengths. Recently quenched galaxies should have younger stellar ages
in the centers; multi-wavelength data with high spatial resolution are required
to reveal the age gradient. Highly dissipative processes should be responsible
for this type of formation history. While determining the mechanisms for
individual galaxies is challenging, some recently quenched galaxies show signs
of gravitational interactions, suggesting that mergers are likely an important
mechanism in triggering the rapid shut-down of star-formation activities at
$z\sim0.8$.
