In its deprotonated form (at neutral pH and higher), most carboxyl groups exists as carboxylates (R-COO-). Since the dibromosuccinate contains two carboxylate groups, it has two negative charges on it and is therefore very soluble in water (a polar solvent). However, at low pH, the carboxylate groups will become protonated and become carboxylic acids (R-COOH). Therefore, when you add HCl to the dibromosuccinate, you get dibromosuccinic acid, a carboxylic acid. Since the carboxylic acid groups are not charged, dibromosuccinic acid is no longer soluble in water and precipitates from the solution.
This principle works for most organic acids. For example, while benzoate ion is soluble in water, benzoic acid is relatively insoluble in water.