Two-dimensional boron, borophene, was realized in recent experiments but still lacks an adequate growth theory for guiding its controlled synthesis. Combining ab initio calculations and experimental characterization, we study edges and growth kinetics of borophene on Ag(111). In equilibrium, the borophene edges are distinctly reconstructed with exceptionally low energies, in contrast to those of other two-dimensional materials. Away from equilibrium, sequential docking of boron feeding species to the reconstructed edges tends to extend the given lattice out of numerous polymorphic structures. Furthermore, each edge can grow via multiple energy pathways of atomic row assembly due to variable boron-boron coordination. These pathways reveal different degrees of anisotropic growth kinetics, shaping borophene into diverse elongated hexagonal islands in agreement with experimental observations in terms of morphology as well as edge orientation and periodicity. These results further suggest that ultrathin borophene ribbons can be grown at low temperature and low boron chemical potential.