To achieve high performance without substantial overheads associated with channel state information (CSI) of ground users, we consider a fixed-beam precoding approach, where a satellite forms multiple fixed-beams without relying on CSI, then select a suitable user set for each beam. Upon this precoding method, we put forth a satellite equipped with massive multiple-input multiple-output (MIMO), by which inter-beam interference is efficiently mitigated by narrowing corresponding beam width. By modeling the ground users' locations via a Poisson point process, we rigorously analyze the achievable performance of the presented multibeam satellite system. In particular, we investigate the asymptotic scaling laws that reveal the interplay between the user density, the number of beams, and the number of antennas. Our analysis offers critical design insights for the multibeam satellite with massive MIMO: i) If the user density scales in power with the number of antennas, the considered precoding can achieve a linear fraction of the optimal rate in the asymptotic regime. ii) A certain additional scaling factor for the user density is needed as the number of beams increases to maintain the asymptotic optimality.