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We investigate the formation and evolution of z = 0 massive compact galaxies (MCGs) in the IllustrisTNG cosmological simulation. We found that, as in observations, MCGs are mainly old (median age ∼10.8 Gyr), have supersolar metallicities (median log  Z /Z⊙ ∼ 0.35), and are α-enhanced (median [α/Fe] ∼ 0.25). The age distribution extends to younger ages, however, and a few MCGs are as young as ∼7 Gyr. In general, MCGs assemble their mass early and accrete low angular momentum gas, significantly increasing their mass while growing their size much slower. A small fraction of MCGs follow another evolutionary path, going through a compaction event, with their sizes shrinking by 40 per cent or more. The accretion of low angular momentum gas leads to enhanced supermassive black hole (SMBH) growth, and MCGs reach the threshold SMBH mass of |$\log M_\mathrm{BH}\sim 10^{8.5}\ \mathrm{ M}_\odot$| – when kinetic active galactic nucleus feedback kicks in and quenches the galaxy – earlier than non-compact galaxies. Comparing MCGs to a sample of median-sized quiescent galaxies matched in effective velocity dispersion, we find that their accretion histories are very different. 71 per cent of MCGs do not merge after quenching compared to 37 per cent of median-sized quiescent galaxies. Moreover, tracing these populations back in time, we find that at least a third of median-sized quiescent galaxies do not have a compact progenitor, underscoring that both dry mergers and progenitor bias effects are responsible for the differences in the kinematics and stellar population properties of MCGs and median-sized quiescent galaxies. [ABSTRACT FROM AUTHOR]