For a typical galvanic cell, let's say Ag/Ag+ and Zn/Zn2+. Electrons will flow from the Zn half cell to the Ag half cell. The reaction is
 + Zn(s) \to Zn^{2+}(aq) + 2Ag(s))
Now, I create a new system with the same reaction. In this new system, I have a zinc rod immersed in Ag+ solution. I notice that the same reaction occurs, with silver leaving the solution and plating on the zinc rod, and the concentration of zinc in the solution increases. I also notice the solution gets warmer.
If we break it down into the two half equations:
 & \to Zn(aq) + 2e^- \\<br />Ag^+(aq) + e^- & \to Ag(s) \\ \end{align*})
You can see that the oxidation of zinc does not involve silver. Zinc in this case simply wants to donate electrons to become more 'stable'. Silver ions do not require zinc to undergo reduction, it simply wants to accept electrons to become more 'stable'. In fact, the thing about redox reactions are they always happen in two half equations, and do not require contact with the other counter-part. Remember that a reaction only requires contact on a molecular level if the reaction requires a collision. In this case, the reaction does not require collision between Zn and Ag+, it simply just needs electrons buzzing around (which can travel via a copper wire). Based on this type of reasoning, you can see how fuel cells operate as well, they simply split a combustion reaction (also redox) into two half equations, and force electrons travel from the fuel end to the oxygen end.
Now, back to galvanic cells. Right now, we have two disconnected cells (no salt bridge, no wires, no connection at all). We have a zinc half cell with Zn(s) wanting to give away electrons, but it cannot give electrons because they have nowhere to go (polarization in the zinc rod, accumulation of -ve charge). Same deal with the silver half cell, that if a reaction is to proceed, it'll cause the silver electrode to become +ve charged, thus polarization.
Now, we electrically connect the two electrodes. Zn still wants to give electrons, now the electrons have somewhere to go, thus they go towards the silver cell. Silver still wish to accept, now there is a source of electrons, reduction occurs and everyone's happy. Except now the solution is becoming polarized because of accumulation of aqueous ions. We now add a salt bridge, and voila, the two reactants are in separate places but still doing the same chemistry. The heat of reaction is carried by the electrons, which can be utilized by electric circuits to transform to useful things instead of heat.