First we have to establish the chemical reaction. As a result of adding Na2CO3 and AgNO3 products formed are NaNO3 and Ag2CO3. NaNO3 is a soluble substance when dissolved in water.
As a general rule, most Ag, Pb and Hg(I) compounds are considered insoluble. Ag2CO3 is insoluble in water hence the reaction given forms a precipitate.