So, we know that:
$$ \ce{NaCl(aq) + AgNO3(aq) -> NaNO3(aq) + AgCl(s)}$$
(In other words, the ions swap and a precipitate forms).
However, both of those compounds seem pretty stable, since $\ce{Ag}$ and $\ce{Na}$ cations both have a 1+ charge, and $\ce{Cl}$ and $\ce{NO3}$ both have a 1- charge. Since they are stable, I would assume no reaction would take place since both ions are happy with the ions they are currently bonded to (sodium and silver, respectively).
However, $\ce{Na}$ is higher on the activity series than $\ce{Ag}$, and we know that these aqueous solutions actually contain dissociated ions, so I assume this has something to do with the double-replacement. But why is this? Why does this reaction take place, even though the original compounds appear stable at first glance?
Edit
I read this question, and it says
Le Châtelier's principle only states that a system previously at equilibrium will want to stay at equilibrium - that is, if we perturb it, it will try to go back to equilibrium.
The above compounds of sodium chloride and silver nitrate appear to be at an initial equilibrium, do they not?
Edit 2
The question isn't why silver chloride is a solid, but rather, why does this reaction take place if the initial two compounds are stable to begin with?