The Belgian cast is also known as the oval cast or the constant tension cast. It is most useful in two situations. First, in windy conditions where the wind is blowing into your casting arm side and second when you are fishing with a double fly setup or maybe with a split shot or heavy nymph and indicator. In both cases, this cast is meant to help keep the fly(s) from smacking you. It is a popular saltwater cast due to the windy conditions that are common.
These two videos do an excellent job of explaining them.
Heavy Fly Casting with Tim Rajeff - Gink and Gasoline
How To Make A Belgian (Oval) Cast - Fly Casting Lessons - ORVIS