Consider a battery electrode as a system. That system cannot create charge (due to conservation of charge). Also, objects generally cannot acquire a net charge. Sure, they can acquire a small net charge due to static electricity or what have you. But generally, in circuit analysis, objects are not allowed to acquire a net charge. Charges can separate and move around, but they cannot be created. If electrons leave an atom, the atom becomes a positively charged cation. Thus, charge is conserved.
So, back to the electrode system. In order to respect conservation of charge, and avoid acquiring net charge, every negative charge that leaves the electrode through the wire must be balanced out by a positive charge leaving the electrode into the electrolyte solution. Likewise, every electron that enters the other electrode must be balanced out by a positive charge entering the electrode through the electrolyte solution.
The purpose of the salt bridge is to allow the positive charges to migrate from one electrolyte bath to the other. There must be a positive ion circuit between the two battery terminals in order for electrons to flow through the external circuit.
OK, an alternate explanation. Consider two systems. Each one is an electrolyte solution with a partially submerged electrode. There is no salt bridge. We add a wire connecting the dry portions of the electrodes (which now makes our two systems into a single system). Let us say that for a brief moment, a tiny current flows in the wire. This current actually imparts a very slight net charge to the electrolyte solutions (equal and opposite). This net charge changes the favorability of the electrochemical reaction which produces current in the battery, and shuts it down. When you add a salt bridge, it allows cations to escape the more positively charged solution, and thus allows current to flow through the wire on a more sustainable basis.
Net charge is self-arresting, because each charge you transfer makes it that much harder to transfer additional charges (because like charges repel each other).