the background limits ‘are the potentials where the cathodic and anodic currents start to flow at a working electrode when it is immersed in a solution containing only an electrolyte added to decrease the solution resistance (a supporting electrolyte).
Bard's book is notoriously complex. Some bright scientists are not good at communication. Richard Feynman was an exception.
All Bard is saying that the background limits define the potential window [see flat line in your figure] where electrolysis does not occur on the working electrode. The background limits are dependent on the electrolyte used, it is also dependent on the solvent and the electrode material as well.
Now keep in mind that modern potential sweep experiments there are three electrodes in a cell. There is a reference electrode, in your case, an Ag/AgBr electrode. Then you have a working electrode, Pt/H+ and then the writer is quiet about the third one, which is called the counter electrode. Most of the time it is understood that it is their. It could be another Pt electrode but not necessarily a hydrogen electrode i.e., nobody is bubbling hydrogen gas over the counter electrode.
Now the reference electrode has no role in the cell. There is no redox reaction taking place at the reference electrode because the circuitry is such that no or very little current passes through the reference electrode. Electrochemists don't care what is happening at the counter electrode. The potential difference is measured between the reference electrode and the working electrode. This is the only purpose of the reference electrode.
When you are sweeping (=continuously changing) the potential of the working electrode with respect to the reference electrode, it can become a cathode or an anode during scanning of the potential. You can scan in any direction. You are right, isolated reduction or oxidation cannot occur. If the working electrode was a cathode at a particular moment, the counter electrode is the anode and vice versa.