J.R. S. answered 12/15/20
Ph.D. University Professor with 10+ years Tutoring Experience
You really only need consider the Zn, Zn2+, Ag and Ag+ to understand what's happening.
The Zn is oxidized and so releases electrons which travel to the cathode(Ag) where the Ag is then reduced. You cannot have Zn2+ and Ag+ present without a counter anion, and hence the presence of the electrolyte.
Zn(s) | Zn2+ || Ag+ | Ag(s)
Because the reduction potential for Ag+ + e- --> Ag(s) is greater than the reduction potential for Zn2+ + 2e- --> Zn(s), the electrons flow from Zn electrode to Ag electrode. Does this makes sense, and does it answer your question? This is what drives Zn to undergo oxidation. It's the difference in reduction potentials between the two metals.