Okay...
When you're shooting video with an HDSLR you have three variables that control your exposure
ISO
Aperture
Shutter Speed
These all control both exposure and have a seperate effect on the image.
ISO - if you raise the number you get more digital gain in your image or "noise" it also increases the exposure
Aperture - When you set it to a lower number and open it up you increase the exposure and earn a shallower Depth of Field, essentially the shallower the DOF the faster things fall out of focus as they move away from the optimal focal plane.
Sutter speed - when you use a lower number you increase the exposure. The shutter speed also controls motion blur in your image. A slower shutter speed has more blur and a faster shutter speed has less blur as things move through the frame.
People like to have control over all of these elements and that's where an ND filter comes in.
Say you're shooting in bright sunlight. Without an ND filter your image is properly exposed at ISO 100, F/16 (aperture), and 1/125 (shutter speed).
Now you have a very busy background and it's distracting. You want to isolate your subject and can only use available light. Naturally you'd opt for a shallower depth of field.
However, when you open up your aperture in order to narrow your DOF your image gets over exposed. You're ISO is already as low as it can go so your only option to properly expose your image is to increase your shutter speed.
Unfortunately this increase in shutter speed makes your footage look very choppy, you get strange artifacts when your subject makes quick movements. When the shutter speed was at 1/125 the image looked much better.
So what you do is slap on an ND filter. This filter darkens the overall image so you can bring that shutter speed back down to 1/125th or something close to that and your shot looks exactly the way you wanted.
Does that make sense?