NGC 4945 is a rather innocuous late-type spiral, except for the energetics of its nucleus. Manifestations of the high level of nuclear activity include very high far-infrared fluxes and a plethora of excited molecules. Using new data obtained with a long-slit near-infrared spectrograph, we deduce the existence of circumnuclear molecular hydrogen which is both spatially and spectrally resolved. The molecular gas is heated by shocks emanating from a nuclear stellar cluster. We find that the flat H I rotation curve valid for the bulk of the galaxy is not appropriate for the velocity distribution of the molecular material in the central kiloparsec. Instead, a massive central star cluster induces Keplerian rotational velocities of 200-250 km s-1 at a galactic radius of 100 parsec, while at distances from the nucleus of 400-500 parsec, velocities of about 100 km s-1 prevail. The spatial and velocity distribution of the Brgamma neutral hydrogen emission line is very similar to the molecular component, but its intensity is remarkably low. No evidence was found for Seyfert 2 nuclear activity as has been argued. Instead, we derive a self-consistent model in which the nuclear stellar cluster is in a post-starburst phase, i.e., without significant numbers of hot main-sequence stars. Mass loss from massive red stars, and supernovae from red supergiant progenitors, dominate the energetics. We propose the name Postburst Infrared Galaxy for objects in this particular evolutionary stage.