Brominated vegetable oil is ordinary oil, such as olive oil, that has had bromine added at various places to make the molecule have the density of the sugar water in soft drinks.
This allows BVO to keep citrus and other oil-based flavorings in solution without separating into an oil layer floating on top of the beverage.
The result is the cloudy solution you see in orange flavored or grapefruit flavored soft drinks.
The chemical formula depicted above is that of trilinolein, a representative component of vegetable oils. Bromine atoms would attach at various places on the molecule to form BVO. The amount of bromine is controlled to produce the desired density.