When AgNO3 is added to the above reaction, Ag combines with the Cl- to form silver chloride, a precipitate. NO3- is a spectator ion and does not react. Why would the addition of silver nitrate stress the equilibrium?

The solubility of AgCl in water decreases.

Explanation:

When we add AgNO₃ we are adding silver ions. Therefore, in addition to the dissociation of AgCl, the following  process also contributes to the total concentration of the common silver ions  in solution:

AgNO₃ (s) → Ag⁺ (aq) + NO₃⁻  (aq)

The solubility equilibrium of AgCl is:

AgCl(s) ⇄ Ag⁺ (aq) + Cl⁻ (aq)

If AgNO₃ is added to a saturated AgCl solution, the increase in [Ag⁺] will make the  ion product greater than the solubility product:

Q = [Ag⁺]₀[ Cl⁻]₀ > Ksp

To reestablish equilibrium, some AgCl will precipitate out of the solution, as Le  Châtelier’s principle would predict, until the ion product is once again equal to Ksp.

The effect of adding a common ion, then, is a decrease in the solubility of the salt  (AgCl) in solution. Note that in this case [Ag⁺] is no longer equal to [Cl⁻] at equilibrium.