The equilibrium that is set up is this,
[Co(H2O)6]2+(aq) + 4Cl–(aq) ⇌ [CoCl4]2-(aq) + 6H2O(l)
and the aqueous solution that is added dropwise is 0.1 M silver nitrate.
If you would like an explanation of what you see, then read on…
…the equilibrium starts out with a purplish color, suggesting that a combination of both the pink and blue species are present in the test tube.
Addition of the silver nitrate solution (effectively a source of silver ions), causes the precipitation of solid silver chloride according to the net, ionic equation,
Ag+(aq) + Cl–(aq) ➔ AgCl(s)
This reaction causes the removal of free chloride ions from the equilibrium system, thus disturbing the equilibrium, and making Q (the reaction quotient) too large. The equilibrium shifts backwards in order to reduce the value of Q, and to bring it back into agreement with the equilibrium constant, K.
As the reaction goes backwards one sees the pink cobalt species begin to dominate, and the formation of the white precipitate of AgCl.
Neat and simple.