The oxidation process either produces positive ions or removes negative ions from the solution at the anode (or it may change one ion to a more positive one), and the reduction process either removes positive ions or produces negative ions in the solution at the cathode. This produces electrically charged solutions, and very quickly stops the process before a measureable number of electrons are transferred.
There must be a path for the ions to move between the two solutions in order for electrons to flow continuously through the wire. This produces an "ion current" within the battery with cations (positively - charged ions) moving from anode to cathode, and anions (negatively - charged ions) moving from the cathode toward the anode.
Now, more the no. of salt bridges, more will be the ion current. And from the standard ohms law equation, $$V=IR$$ , V being directly proportional to I (Current), will increase if the ion current increases.