To put it simply, a matte box is the square thing that you see on the end of movie/video cameras. They have three uses.
1. It prevents lens flare
Unless you want it in your shot, a matte box is designed to cut out lens flare. Usually, a matte box will have adjustable fins on the top and sides. This is its main purpose.
2. It holds filters
More expensive matte boxes allow you to slot filters, like neutral density filters, in front of your lens. This allows you to replace filters faster than the ones that screw directly onto your lenses. Obviously this doesn't really work well if you're using a zoom lens that extends in and out since the matte box is usually fixed to rails beneath the camera.
3. It looks pro
You put a matte box in front of your camera and it automatically looks 100% cooler. Yeah, I know this isn't a proper reason for having one but it does make a DSLR look less like a photography camera and more like a proper movie camera. Image is everything and you should do everything you can to disguise your DSLR when you take it to a video shoot.