Nonlinear electron trapping physics governs the onset and saturation of stimulated Raman scattering (SRS) in laser beams with many speckles. Hot electrons from intense speckles, produced during SRS daughter electron-plasma-wave bowing and filamentation, seed and enhance the growth of SRS in neighboring speckles by reducing Landau damping. Trapping-induced nonlinearity and speckle interaction through transport of hot electrons and back- and sidescattered SRS waves enable the system of speckles to self-organize and exhibit coherent, sub-ps SRS bursts with more than 100% instantaneous reflectivity, consistent with a SRS transverse coherence width much larger than a speckle width.