The progenitors of astronomical transients are linked to a specific stellar population and galactic environment, and observing their host galaxies hence constrains the physical nature of the transient itself. Here, we use imaging from the Hubble Space Telescope, and spatially resolved, medium-resolution spectroscopy from the Very Large Telescope obtained with X-shooter and MUSE to study the host of the very luminous transient ASASSN-15lh. The dominant stellar population at the transient site is old (around 1 to 2 Gyr) without signs of recent star formation. We also detect emission from ionized gas, originating from three different, time invariable, narrow components of collisionally excited metal and Balmer lines. The ratios of emission lines in the Baldwin-Phillips-Terlevich diagnostic diagram indicate that the ionization source is a weak active galactic nucleus with a black hole mass of M. = 5(-3)(+8) x 10(8) M-circle dot, derived through the M.-sigma relation. The narrow line components show spatial and velocity off sets on scales of 1 kpc and 500 km s(-1), respectively; these off sets are best explained by gas kinematics in the narrow-line region. The location of the central component, which we argue is also the position of the supermassive black hole, aligns with that of the transient within an uncertainty of 170 pc. Using this positional coincidence as well as other similarities with the hosts of tidal disruption events, we strengthen the argument that the transient emission observed as ASASSN-15lh is related to the disruption of a star around a supermassive black hole, most probably spinning with a Kerr parameter a. greater than or similar to 0.5.