Why is insoluble salt washed with distilled water

For illustration, you want to prepare a pure sample of CaCO3. This is often done in the pharmaceutical industry for the production of CaCO3 filler used in tablets and antacids. 
You can do this by mixing solutions of CaCl2 and Na2CO3 
CaCl2(aq) + Na2CO3(aq) →CaCO3(s) + 2NaCl(aq) 
The CaCO3 preciitates as a very fine precipitate. The NaCl solution is drained away. Any NaCl adhering to the CaCO3 is removed by further washing of the precipitate and recovering the CaCO3 by filtration. For this to be effective, the NaCl has to be readily soluble in water, which it is. 
You cannot have reactions occuring where the two reactants are insoluble in the water, and it is unusual, but not unknown, to produce two precipitates from the single reaction.

Hope this helps!