Rembrandt was simply painting window light. That's how people look in interiors in the absence of electricity, nearby reflectors, or large numbers of windows. Turn off the lights, go over to the window, you got Rembrandt light. We've all used it if only by default or as a last recourse. Too common a look for many of Rembrandt's contemporaries, most upper class photo subjects would rather have been seen in more luminous environments, with carried the implication of wealth and the extravagant numbers of expensive glass covered windows that could bring.
Rembrandt actually got the look from Caravaggio, who was a few decades older. Caravaggio was an art bad boy who painted relatively shocking subjects in really hard, unattractive, directional light, and let's just say he sometimes challenged conservative family values. Rembrandt softened things up a bit and chose less controversial subjects. Rembrandt was a north light type of guy, whereas Caravaggio often had hard, direct sunlight pouring through his window. But both very photographic, and Caravaggio in particular had a very sharp lens indeed.
