There is no explanation for this, same as there is no real explanation for why energy is conserved or why closed systems have non-decreasing entropy. As others have pointed out, you can show correspondence to Newtonian mechanics under some assumptions, but the Lagrangian approach is applicable to a wide variety of areas in physics - classical mechanics, optics, quantum mechanics, quantum field theory, etc.

The universe has these weird laws, and for now, all we can do is accept them as is. But hopefully, in the future, someone will figure out deeper and simpler principles.