We present Multi Unit Spectroscopic Explorer (MUSE) observations in the core of the Hubble Frontier Fields (HFF) galaxy cluster MACS J1149.5+2223, where the first magnified and spatially resolved multiple images of supernova (SN) "Refsdal" at redshift 1.489 were detected. Thanks to a Director's Discretionary Time program with the Very Large Telescope and the extraordinary efficiency of MUSE, we measure 117 secure redshifts with just 4.8 hr of total integration time on a single 1 arcmin2 target pointing. We spectroscopically confirm 68 galaxy cluster members, with redshift values ranging from 0.5272 to 0.5660, and 18 multiple images belonging to seven background, lensed sources distributed in redshifts between 1.240 and 3.703. Starting from the combination of our catalog with those obtained from extensive spectroscopic and photometric campaigns using the Hubble Space Telescope (HST), we select a sample of 300 (164 spectroscopic and 136 photometric) cluster members, within approximately 500 kpc from the brightest cluster galaxy, and a set of 88 reliable multiple images associated with 10 different background source galaxies and 18 distinct knots in the spiral galaxy hosting SN "Refsdal." We exploit this valuable information to build six detailed strong-lensing models, the best of which reproduces the observed positions of the multiple images with an rms offset of only 0.″26. We use these models to quantify the statistical and systematic errors on the predicted values of magnification and time delay of the next emerging image of SN "Refsdal." We find that its peak luminosity should occur between 2016 March and June and should be approximately 20% fainter than the dimmest (S4) of the previously detected images but above the detection limit of the planned HST/WFC3 follow-up. We present our two-dimensional reconstruction of the cluster mass density distribution and of the SN "Refsdal" host galaxy surface brightness distribution. We outline the road map toward even better strong-lensing models with a synergetic MUSE and HST effort.