Why adjust the shutter speed? The answer lies in your subject. For me, it's kids. They move too quickly and unless you adjust the shutter speed you're going to get a lot of parents standing still with blurry kids running around. It's also great for sports, birds flapping their wings, and other fast moving things that I haven't shot yet.
Once you've switched to Manual mode you'll be able to adjust the shutter so it fires very quickly. Try to keep it 1/200th of a second or faster if you can (if your kids are running around – to up to 1/500th or more). And depending on your camera you probably have the ability to set it take multiple shots at once by holding down the shutter button. It also gives a nice sound with those photos being snapped all at once.
Conversely you can adjust the shutter speed to fire very slowly. This is great for darkly lit scenes or even astronomy photography. This is an area I want to learn more about.
Kids don't respond well to being told to stand still while Daddy takes a picture. When you increase the shutter speed you can get out of the way and let kids be kids, capturing them in their fast moving native environment. Like cheetahs!