Still's cameras have been able to shoot video for a while but it has always been atrocious quality and certainly not anything you would think of using for making a film but since the mad scientists at Canon decided it would be possible to pump out 30 frames of 1920x1080 video every second from their giant full frame sensor, everything has changed.
So the puzzle is, why would a filmmaker be interested in using a stills camera to make his film. It is that massive sensor that is at the centre of this puzzle. The Canon 5d literally has the biggest sensor of any digital HD video camera (and that includes such pricey items as the Red One and the Sony F35). This gives the 5D two main advantages, firstly it gives it an obscenely shallow depth of field, secondly it makes the camera fairly sensitive to light which means you can shot with less light (possibly even streetlights at night) than would be needed with other cameras.
Depth of field (DOF) refers to the distance between the nearest and the furthest point from the camera that will be in focus. A major criticism of HD cameras is that they have too deep a DOF, in other words too much is in focus, and this stops them from looking "filmic" like the image above. DOF is dependent on 3 factors, the focal distance of the lens, the openness of the aperture (called the F-stop) and the size of imagining plane. In the case of focal length a wide angle lens will have a very deep DOF (see the deep focus cinematography in classic films like Citizen Kane) whilst telephoto lenses will have a shallow depth of field. With the aperture or iris, the more you open it (make the F-stop lower) the shallower the DOF so if you are going for a shallow DOF then its a good idea to use ND filters to limit the amount of light entering the camera before you close the aperture any. Lastly the imaging plane affects DOF. Imaging plane means the plane that the image is being projected onto by the lens. In a film camera this would be the film itself. In a digital camera, this would be the sensor.