Supercontinuum generation can be achieved in the continuous-wave regime with a few watts of pump power launched into kilometer-long fibers. High power spectral density broadband light sources can be obtained in this way. Using a generalized nonlinear Schrödinger equation model and an ensemble averaging procedure that takes into account the partially-coherent nature of the pump laser, we fully explain for the first time the spectral broadening mechanisms underlying this process. Our simulations and experiments confirm that continuous-wave supercontinuum generation involve Raman soliton dynamics and dispersive waves in a way akin to pulsed supercontinua. The Raman solitons are however generated with a wide distribution of parameters because they originate from the random phase and intensity fluctuations associated with the pump incoherence. This soliton distribution is averaged out by experimental measurements, which explains the remarkable smoothness of experimental continuous-wave supercontinuum spectra.