Colloidal nanoplatelets are atomically flat, quasi-two-dimensional sheets of semiconductor that can exhibit efficient, spectrally pure fluorescence. Despite intense interest in their properties, the mechanism behind their highly anisotropic shape and precise atomic-scale thickness remains unclear, and even counter-intuitive for commonly studied nanoplatelets that arise from isotropic crystal structures (such as zincblende CdSe and lead halide perovskites). Here we show that an intrinsic instability in growth kinetics can lead to such highly anisotropic shapes. By combining experimental results on the synthesis of CdSe nanoplatelets with theory predicting enhanced growth on narrow surface facets, we develop a model that explains nanoplatelet formation as well as observed dependencies on time and temperature. Based on standard concepts of volume, surface and edge energies, the resulting growth instability criterion can be directly applied to other crystalline materials. Thus, knowledge of this previously unknown mechanism for controlling shape at the nanoscale can lead to broader libraries of quasi-two-dimensional materials.