If I am correct, I believe it is the common ion effect. Props to you firecloak
"If both
sodium acetate and
acetic acid are dissolved in the same solution they both dissociate and ionize to produce
acetate ions. Sodium acetate is a
strong electrolyte so it dissociates completely in solution. Acetic acid is a
weak acid so it only ionizes slightly. According to
Le Chatelier's principle, the addition of acetate ions from sodium acetate will suppress the ionization of acetic acid and shift its
equilibrium to the left. Thus the percent dissociation of the acetic acid will decrease and the pH of the solution will increase. The ionization of an acid or a base is limited by the presence of its conjugate base or acid." - Wikipedia: Common-ion effect '
NaC2H3O2(s) → Na+(aq) + C2H3O2−(aq)
HC2H3O2(l) ↔ H+(aq) + C2H3O2−(aq)
Furthermore, in your case, "The solubility of a sparingly soluble salt is reduced in a solution that contains an ion in common with that salt. For instance, the solubility of AgCl(Silver Chloride) in water is suppressed if a solution of NaCl(Sodium Chloride) is added to the mixture."-Once again...Wikipedia.
So in your case, NaI is a strong electrolyte and will dissociate completely. Where as AgI is a weaker electrolyte. Since all all the NaI will dissociate completely and the AgI will dissociate partially, the equilbrium will shift to the left, as stated in the article above.
Hope that helps.