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Active matter systems are inherently out of equilibrium and break the detailed balance (DB) at the microscopic scale, exhibiting vital collective phenomena such as motility-induced phase separation (MIPS). Here, we introduce a coarse-grained mapping method to probe DB breaking in the density-energy phase space, which allows us to reveal the dynamic and thermodynamic origins of MIPS based on nonequilibrium potential and flux landscape theory. Hallmarks of nonequilibrium properties are manifested by identifying the visible probability flux in the coarse-grained phase space. Remarkably, the flux for the system with the activity lower than the MIPS threshold tends to ``tear up" the single potential well of the uniform-density phase to create two wells of phases with different densities, presenting directly that the nonequilibrium flux is the dynamic origin of MIPS. Moreover, we find that the obtained entropy production rate (EPR) of the system undergoes a transition from nearly independent of activity to increasing proportionally as activity increases after the single well is "teared up". The transition of EPR's scaling behavior might provide a hint of the thermodynamic origin of MIPS in the coarse-grained space. Our findings propose a new route to explore the nonequilibrium nature of active systems, and provide new insights into dynamic and thermodynamic properties of MIPS.