Organic-inorganic hybrid Ruddlesden-Popper perovskites (HRPPs) have gained much attention for optoelectronic applications due to their high moisture resistance, good processability under ambient conditions, and long functional lifetimes. Recent success in isolating molecularly thin hybrid perovskite nanosheets and their intriguing edge phenomena have raised the need for understanding the role of edges and the properties that dictate their fundamental behaviors. In this work, we perform a prototypical study on the edge effects in ultrathin hybrid perovskites by considering monolayer (BA)2PbI4 as a representative system. On the basis of first-principles simulations of nanoribbon models, we show that in addition to significant distortions of the octahedra network at the edges, strong edge stresses are also present in the material. Structural instabilities that arise from the edge stress could drive the relaxation process and dominate the morphological response of edges in practice. A clear downward shift of the bands at the narrower ribbons, as indicative of the edge effect, facilitates the separation of photoexcited carriers (electrons move toward the edge and holes move toward the interior part of the nanosheet). Moreover, the desorption energy of the organic molecule can also be much lower at the free edges, making it easier for functionalization and/or substitution events to take place. The findings reported in this work elucidate the underlying mechanisms responsible for edge states in HRPPs and will be important in guiding the rational design and development of high-performance layer-edge devices.