What is the use of silver ion?
The step in question here is the so-called Wolff Rearrangement.

In this rearrangement, silver in the form of for example $\ce{Ag2O}$ is used as a catalyst. A catalyst facilitates a reaction (lowering the activation energy), but it comes out of the reaction unchanged (which doesn't mean it's not changed during the reaction, it's just changed back at the end ;))

So why silver? Well, transition metals are good catalysts most of the time for different reasons:


They have vacant d-orbitals which leads to
An easy tendency to exhibit variable oxidation states.
They can form reaction intermediates with reactants.
The presence of defects in their crystal lattices.


(I conveniently copied this list from a quora post, because im lazy)

What this essentially means is that due to their electronic structure, transition metals are often able to bind reactants to their surface and activate them for example. There are quite some catalyst reactions that we don't even understand today, but sometimes it's as easy as this 



And why silver? Well probably Wolff just tried a couple of transition metal catalysts and chose the one with the best effect/cost ratio! Theoretical expectations don't always meet reality so often you just have to try and find the right reactants/catalysts.