I'm very confused about spectator ions. I understand the definition of a spectator ion but I am not clear on how you decipher which cations and anions are the spectator ions.
For instance:
KCl+AgNO3--->AgCl+KNO3

I understand that the potassium chloride dissociate into K+ and Cl- and the silver nitrate dissociates into Ag+ and NO3-

My textbook tells me that through metathesis reaction the product becomes AgCl and that the potassium and chloride are the spectator ions but I don't understand why the potassium and chloride ions don't react with one another and join. My textbook says that they are unchanged on both sides of the equation but I don't understand why that is. Any help at all would be greatly appreciated. Thanks.

Because they are both disassociated in solution.

K^+ ( the cation on both sides of the equation )

and

NO3^- ( the polyatomic anion on both sides of the equation )

In solution these ions are always surrounded by polar water. The AgCl precipitates out of solution as a solid.