We use the EAGLE suite of cosmological hydrodynamical simulations to study how the HI content of present-day galaxies depends on their environment. We show that EAGLE reproduces observed HI mass-environment trends very well, while semi-analytic models typically overpredict the average HI masses in dense environments. The environmental processes act primarily as an on/off switch for the HI content of satellites with stellar mass Mstar>10^9 Msun. At a fixed Mstar, the fraction of HI-depleted satellites increases with increasing host halo mass M200 in response to stronger environmental effects, while at a fixed M200 it decreases with increasing satellite Mstar as the gas is confined by deeper gravitational potentials. HI-depleted satellites reside mostly, but not exclusively, within the virial radius r200 of their host halo. We investigate the origin of these trends by focussing on three environmental mechanisms: ram pressure stripping by the intra-group medium, tidal stripping by the host halo, and satellite-satellite encounters. By tracking back in time the evolution of the HI-depleted satellites, we find that the most common cause of HI removal is satellite encounters. The timescale for HI removal is typically less than 0.5 Gyr. Tidal stripping occurs in halos of M200<10^{14} Msun within half r200, while the other processes act also in more massive halos, generally within r200. Conversely, we find that ram pressure stripping is the most common mechanism that disturbs the HI morphology of galaxies at redshift z=0. This implies that HI removal due to satellite-satellite interactions occurs on shorter timescales than the other processes.
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