Why doesn't a reaction occur when thiosulfate ions are added to solid silver iodide, but does occur when added to silver bromide?What role does the equilibrium concentration of silver play in this phenomenon?

AgBr has a Ksp.

AgI has a Ksp.
Ag(S2O3)2^3- is a complex ion with Kf.
Look up those values.

AgI is so much more insoluble that it will not dissolve in Na2S2O3 to form the complex ion. AgBr will dissolve to form Ag(S2O3)2^3-