We present results from a survey of galaxies in the fields of six z ≥ 3 damped Lyman α (Lyα) systems (DLAs) using the Multi Unit Spectroscopic Explorer (MUSE) at the Very Large Telescope (VLT). We report a high detection rate of up to {≈ } 80{{ per cent}} of galaxies within 1000 km s^-1 from DLAs and with impact parameters between 25 and 280 kpc. In particular, we discovered five high-confidence Lyα emitters associated with three DLAs, plus up to nine additional detections across five of the six fields. The majority of the detections are at relatively large impact parameters (>50 kpc) with two detections being plausible host galaxies. Among our detections, we report four galaxies associated with the most metal-poor DLA in our sample (Z/Z_⊙ = -2.33 ± 0.22), which trace an overdense structure resembling a filament. By comparing our detections with predictions from the Evolution and Assembly of GaLaxies and their Environments (EAGLE) cosmological simulations and a semi-analytic model designed to reproduce the observed bias of DLAs at z > 2, we conclude that our observations are consistent with a scenario in which a significant fraction of DLAs trace the neutral regions within haloes with a characteristic mass of M_h ≈ 10^{11}-10^{12} M_\odot, in agreement with the inference made from the large-scale clustering of DLAs. We finally show how larger surveys targeting ≈25 absorbers have the potential of constraining the characteristic masses of haloes hosting high-redshift DLAs with sufficient accuracy to discriminate between different models.