Liquid crystal elastomers (LCE)polymer networks constructed with liquid crystal unitsare functional materials characterized by a pronounced coupling between elastic strain and liquid crystalline orientational ordering. When prepared by polymerization and cross-linking in the isotropic phase, and then cooled (isotropic genesis), a polydomain LCE presents an extraordinary supersoft elastic behavior under unidirectional pulling, with a plateau-like low elastic modulus region in the stressstrain curve, before turning into a monodomain LCE where a standard elastic resistance is recovered. The physical mechanism of this behavior is of great interest, but still mysterious. This work investigates its origin, reproducing the experiment and succeeding to observe the plateau-like stressstrain behavior by means of large-scale molecular iso-stress Monte Carlo simulations. The results not only show that the basic molecular mechanism behind supersoft elasticity hinges on local domain rotation and growth but also suggest that this is assisted by a newly proposed dissipation of the elastic energy stored in the defects created by the isotropic genesis process, explaining also the irreversibility observed in main-chain LCE.