The silver cation is a unique ion, different from all other silver ions. It slides chaotically in the water, electrostatically unfulfilled. It entered the solution with a nitrate anion, which is similarly lost.
But after a time, the silver cation found a bromide anion, which resolved its tension. They fall out of solution.
Things are ideal: There is a silver cation for every bromide anion. Things work out for all the silver ions and all the bromide ions: there are none left in solution.