ND filters are used by amateurs and pros alike for both video and still photography - but what do they actually do, and how do they make photos and video better?
The function of an ND filter is simple: it blocks a percentage of light from entering the camera. It may seem counterintuitive that a camera that works by absorbing and measuring light can actually work better by having some light blocked from going into it, but there are some good reasons why you might want to do that.
But first, because there are many misconceptions about what they do, let's first talk a bit about what ND filters do NOT do:
Give you better colors: They are called "neutral" density filters for a reason - they are supposed to be neutral. A good quality ND filter should not alter that balance of color, such as making it more blue, or green (a "color cast").
Make the image sharper: curved lenses can affect the sharpness (i.e. focus) of the image, but ND filters are usually not curved. ND filters should not affect the focus of the image in any way.
Give you better dynamic range: Generally speaking, an ND filter doesn't give you better dynamic range (in fact if used improperly, it can decrease it). The one instance where it can help with dynamic range is if your scene is so bright that the camera clips the whites by overloading the sensor, losing a lot of detail in the whites (and losing dynamic range). To correct this, an ND filter can block some of the light to recover the lost dynamic rage.
Stop glare: Glare is caused by reflections, and although an ND filter blocks light, it does not reduce the glare. In fact, some ND filters can increase glare because it acts as an additional parallel piece of glass, allowing light to bounce around between the filter and the camera lens, creating glare. Camera Butter ND filters are coated on the inside and outside to reduce these types of reflections.
(I even see some of the above claims made in marketing material by companies that sell ND filters!)
Now that we know what neutral density filters don't do, let's explore how they can be useful when filming video:
- Remove jello
- Add motion blur
- Make objects disappear
- Fix overexposure
- Help you to use a fixed shutter speed
- Protect your lens
Remove Jello
Film cameras and some high end digital video cameras have a global shutter - in other words the shutter opens all at once, letting the entire image hit the sensor at the same time. CMOS-based cameras like the GoPro and other action cameras use a rolling shutter - where the image is measured sequentially in rows. 
One of the problems with rolling shutters is that by the time the capture process reaches the bottom of the frame, some time has passed. If your camera has a lot of movement or vibration, the image that it's measuring by the end of the capture is not the same image that it started measuring at the top - because it's moved. What you can end up with is a rolling, wavy effect called "jello". One common example is the strange bending of propellers when filmed with rolling shutter camera. Here is another great example.
An ND filter reduces the amount of light getting into the camera, and this will force the camera (or the operator) to open up the shutter for longer to capture enough light to properly expose the image. Because the shutter is open longer, the entire image is blurred just enough to "smear out" the jello effect, but no so much to make the image look worse. In fact, the little bit of blur can actually make your footage better (see next section).
Add motion blur
If you consider old stop-motion filmmaking (e.g. claymation). Each scene is created and makes up a single frame. The frames are played one after another to simulate motion. But there was always a problem: Because each frame was a perfectly still image, with no motion blur, the film had a choppy and artificial.
The same thing happens with footage taken in bright daylight (which requires a fast shutter speed) - the footage has no motion blur, and therefore can look choppy/sterile/artificial and a bit strange to the eyes.
Hollywood movies usually incorporate some amount of motion blur into movement - it makes it seem more natural (and exciting) because our eyes normally see motion with some amount of blur.
By using an ND filter, it adds some of that motion blur to our action camera footage- this makes it look more natural, smooth, more "real" and more cinematic because it looks more like a movie looks, and more like what our eyes are accustomed to seeing in real life. It just looks more "professional".
Make objects disappear
Huh? really? If an object is moving quickly enough and your shutter is open long enough, all that will be left of the fast moving object is a little bit of smear. This effect is used by still photographers to remove crowds from outdoor locations. Here's an example of using an ND filter to almost completely remove propellers from drone footage:
Fix overexposure
The main way that your action camera tries to compensate for bright light is by raising the shutter speed - the less time that the shutter is open, the less light that cumulatively reaches the sensor. Cameras have an upper limit to how fast the shutter can work, and if it's really bright, the shutter might not be able to function at a speed fast enough to avoid letting too much light in. In this case, your image ends up being overexposed:
When this happens, your image loses contrast, and can be washed out. It's also possible that the white balance and color algorithms in the camera will not work properly, and you'll end up with unexpected results. Further to that, action camera sensors have a "sweet spot" - a brightness level at which they work best. So even if you can manage to correctly expose your video, you might not be getting an optimum picture quality.
An ND filter can block some of the light making its way into your camera to get it to a manageable level.
Help you to use a fixed shutter speed
One of the things that separates amateur video from pro video is a consistent exposure level. If you use auto exposure, then the camera is continually adjusting the exposure to match the image - this means the the picture gets brighter and darker based on what it's pointing at. Most pro video does not do this (or at least it does it in a very controlled way), so if you want to step up your video quality you would want to try locking your shutter at a certain speed.
The problem is that some cameras do not offer very many manual shutter speeds (the GoPro Session is a great example), and so you may not be able to set your shutter speed high enough to properly expose the picture. In this case, an ND filter can bring the light level down far enough that you will almost certainly be able to find a shutter speed that works.
Protect your lens
An ND filter in front of your camera lens adds just an extra level of protection. Sure, you can break a filter if your camera is involved in some sort of impact or crash, but even then it's acting as a sacrificial piece of glass so your camera lens stays safe!
Now go out an make some awesome video!
Hopefully this has turned you on to some of the many things that ND filters can (and can't) do, how they can solve some problems and how they can make your footage look better and more professional/cinematic. You can see some more of these examples and a more detailed explanation as part of a longer video here.