Because, in a Double Displacement Reaction, reactants mutually swap their ions in two compounds to construct the new products. Examples :
Here, for example, sodium sulphate and barium chloride are the two compounds. Sodium sulphate reacts with barium chloride by mutually swapping their ions to construct sodium chloride and barium sulphate.
In the spot of sodium sulphate, sodium chloride is constructed and in the place of barium chloride, barium sulphate is constructed. In this reaction, chlorine and sulphate are the ions where the ions get transported to the two compounds by mutually swapping their ions in two compounds.
Hence, this type of reaction is named a double displacement reaction.