The process of frictional rupture, i.e., the failure of frictional systems, abounds in the technological and natural world around us, ranging from squealing car brake pads to earthquakes along geological faults. A general framework for understanding and interpreting frictional rupture commonly involves an analogy to ordinary crack propagation, with far-reaching implications for various disciplines from engineering tribology to geophysics. An important feature of the analogy to cracks is the existence of a reduction in the stress-bearing capacity of the ruptured interface, i.e., of a drop from the applied stress, realized far ahead of a propagating rupture, to the residual stress left behind it. Yet, how and under what conditions such finite and well-defined stress drops emerge from basic physics are not well understood. Here, we show that for a rapid rupture a stress drop is directly related to wave radiation from the frictional interface to the bodies surrounding it and to long-range bulk elastodynamics and not exclusively to the physics of the contact interface. Furthermore, we show that the emergence of a stress drop is a transient effect, affected by the wave travel time in finite systems and by the decay of long-range elastic interactions. Finally, we supplement our results for rapid rupture with predictions for a slow rupture. All of the theoretical predictions are supported by available experimental data and by extensive computations. Our findings elucidate the origin of stress drops in frictional rupture; i.e., they offer a comprehensive and fundamental understanding of why, how, and to what extent frictional rupture might be viewed as an ordinary fracture process.