We characterized the kinematics, morphology, and stellar population (SP) properties of a sample of massive compact quiescent galaxies (MCGs, 10<~logM*/M_{sun}_<~11 and re~1-3kpc) in the MaNGA Survey, with the goal of constraining their formation, assembly history, and assessing their relation with non-compact quiescent galaxies. We compared their properties with those of a control sample of median-sized quiescent galaxies (re~4-8kpc) with similar effective velocity dispersions. MCGs have elevated rotational support, as evidenced by a strong anticorrelation between the Gauss-Hermite moment h3 and V/{sigma}. In contrast, 30 per cent of control sample galaxies (CSGs) are slow rotators, and fast-rotating CSGs generally show a weak h3-V/{sigma} anticorrelation. MCGs and CSGs have similar ages, but MCGs are more metal-rich and {alpha}-enhanced. Both MCGs and CSGs have shallow negative metallicity gradients and flat [{alpha}/Fe] gradients. On average, MCGs and CSGs have flat age gradients, but CSGs have a significantly larger dispersion of gradient values. The kinematics and SP properties of MCGs suggest that they experienced highly dissipative gas-rich events, such as mergers, followed by an intense, short, and centrally concentrated burst of star formation, between 4 and 10Gyr ago (z~0.4-2), and had a quiet accretion history since then. This sequence of events might be analogous to, although less extreme than, the compaction events that formed compact quiescent galaxies at z~2. The small sizes of MCGs, and the high efficiency and short duration of their last star formation episode suggest that they are descendants of compact post-starburst galaxies.