Variability selection has been proposed as a powerful tool for identifying both low-luminosity active galactic nuclei (AGN) and those with unusual spectral energy distributions. However, a systematic study of sources selected in such a way has been lacking. In this paper, we present the multiwavelength properties of the variability-selected AGN in GOODS-South. We demonstrate that variability selection indeed reliably identifies AGN, predominantly of low luminosity. We find contamination from stars as well as a very small sample of sources that show no sign of AGN activity, their number is consistent with the expected false positive rate. We also study the host galaxies and environments of the AGN in the sample. Disturbed host morphologies are relatively common. The host galaxies span a wide range in the level of ongoing star formation. However, massive starbursts are only present in the hosts of the most luminous AGN in the sample. There is no clear environmental preference for the AGN sample in general but we find that the most luminous AGN on average avoid dense regions while some low-luminosity AGN hosted by late-type galaxies are found near the centres of groups. AGN in our sample have closer nearest neighbours than the general galaxy population. We find no indications that major mergers are a dominant triggering process for the moderate- to low-luminosity AGN in this sample. The environments and host galaxy properties instead suggest secular processes, in particular tidal processes at first passage and minor mergers, as likely triggers for the objects studied. This study demonstrates the strength of variability selection for AGN and gives first hints at possibly triggering mechanisms for high-redshift low-luminosity AGN.