A light meter is a device used to measure the amount of light. In photography, a light meter is often used to determine the proper exposure for a photograph.
Now why do you think that managing exposure is not needed in Digital Cinematography ?
Photography or Cinematography, Analog or Digital, it's all about managing the exposure.
In plainest terms, Exposure just means how much light is exposed to the picture and most pictures are either under exposed of over exposed. Under exposed means having a dark picture and over exposed means having more light than required.
Good photography or cinematography is all about having just the right amount of light in your picture.
Most of the techniques in photography/cinematography deal with this very thing. Lens Filters do this very job of managing the exposure and this is applicable to both Digital and analog techniques.
Check any photography tutorial (analog/digital), they will tell you about how to manage the exposure.
when something is captured by a camera it goes to the sensor and that's where it is determined how the picture is going to be, depending on its exposure. It's a different story that whether The image or video goes to a film or an SD card.
Summing that up, what I want to say is Digital or Analog they all need to handle exposure. This is irrespective of whether the final output is going to a film or a digital media. And Light meter only tells the current amount of exposure, hence helping the folks to determine how much more they need to tweak the exposure.
P.S there are several techniques to tweak the exposure which are bith pre and post capture.
A few links :
Exposure : Please go through the manual exposure part. It tells about light meter.
they could judge how much light is needed without exactly measuring
the amount of light on screen (by looking at the scene once lit or
watching the camera's output monitor).
This is really not so easy. You can't estimate the right amount of light just by looking at the output monitor. A huge amount of precision is required to determine the right exposure and we definitely need an instrument to help us determine that.