Think of it like mixing two clear liquids that suddenly produce 'snow' at the bottom of the beaker; the 'snow' is the insoluble substance that refuses to dissolve in the water.

This is a classic example of a precipitation reaction where ions exchange partners in an aqueous solution. The barium ions ($Ba^{2+}$) from the barium chloride and the sulphate ions ($SO_4^{2-}$) from the sodium sulphate combine to form the highly insoluble compound $BaSO_4$. The driving force of the reaction is the formation of this insoluble solid, which effectively removes ions from the solution.

Remember that in double displacement reactions, always look for the formation of an insoluble product (precipitate) or a gas to predict the reaction outcome.