Colloidal nanoplatelets - quasi-two-dimensional sheets of semiconductor exhibiting efficient, spectrally pure fluorescence - form when liquid-phase syntheses of spherical quantum dots are modified. Despite intense interest in their properties, the mechanism behind their anisotropic shape and precise atomic-scale thickness remains unclear, and even counterintuitive when their crystal structure is isotropic. One commonly accepted explanation is that nanoclusters nucleate within molecular templates and then fuse. Here, we test this mechanism for zincblende nanoplatelets and show that they form instead due to an intrinsic instability in growth kinetics. We synthesize CdSe and CdS1-xSex nanoplatelets in template- and solvent-free isotropic melts containing only cadmium carboxylate and chalcogen, a finding incompatible with previous explanations. Our model, based on theoretical results showing enhanced growth on narrow surface facets, rationalizes nanoplatelet formation and experimental dependencies on temperature, time, and carboxylate length. Such understanding should lead to improved syntheses, controlled growth on surfaces, and broader libraries of nanoplatelet materials.