The discovery of quasars a few hundred megayears after the Big Bang represents a major challenge to our understanding of black holes as well as galaxy formation and evolution. Quasars' luminosity is produced by extreme gas accretion onto black holes, which have already reached masses of M_BH_>10^9^M_{sun}_ by z~6. Simultaneously, their host galaxies form hundreds of stars per year, using up gas in the process. To understand which environments are able to sustain the rapid formation of these extreme sources, we started a Very Large Telescope/Multi-Unit Spectroscopic Explorer (MUSE) effort aimed at characterizing the surroundings of a sample of 5.7<z<6.6 quasars, which we have dubbed the Reionization Epoch QUasar InvEstigation with MUSE (REQUIEM) survey. We here present results of our searches for extended Ly{alpha} halos around the first 31 targets observed as part of this program. Reaching 5{sigma} surface brightness limits of 0.1-1.1x10^-17^erg/s/cm^2^/arcsec^2^ over a 1arcsec^2^ aperture, we were able to unveil the presence of 12 Ly{alpha} nebulae, eight of which are newly discovered. The detected nebulae show a variety of emission properties and morphologies with luminosities ranging from 8x10^42^ to 2x10^44^erg/s, FWHMs between 300 and 1700km/s, sizes <30pkpc, and redshifts consistent with those of the quasar host galaxies. As the first statistical and homogeneous investigation of the circumgalactic medium of massive galaxies at the end of the reionization epoch, the REQUIEM survey enables the study of the evolution of the cool gas surrounding quasars in the first 3Gyr of the universe. A comparison with the extended Ly{alpha} emission observed around bright (M_1450_<~-25mag) quasars at intermediate redshift indicates little variations on the properties of the cool gas from z~6 to z~3, followed by a decline in the average surface brightness down to z~2.