Let say the cheap metal rod used is Iron.
If you just immerse the iron rod into the gold solution, the displaced gold ions will form a coating on the unreacted iron, thus preventing furthur reaction. However, if you use the setup I had proposed, the displaced gold ions will deposit on the gold rod instead. In this way, the purity of your product (solid gold) is assured.
Fe (s) -> Fe2+ (aq) + 2e
The solvation of Iron is actually an oxidation process, which is described by the equation above. The electrons are then transferred from the Iron rod to the Gold rod via the wire. This makes the Gold rod negatively charged, thus attracting cations to the Gold rod.
Au+ (aq) + e -> Au (s)
The aqueous cation which is most susceptible to reduction is in fact the aqueous Au(I) cation in the solution. The Au(I) cation readily accepts the electron from the negatively charged gold rod and reduces to solid gold, thus depositing on the gold rod.
The electromotive force (EMF) driving the flow of electrons from the Iron rod to the Gold rod comes from the potential difference between Gold and Iron.
The overall reaction for the process is:
Fe (s) + 2Au+ (aq) -> Fe2+ (aq) + 2 Au (s)