Antimalarials, fever, age (of the RBC), other drugs, and a host of other things increase oxidative stress in RBCs (which results in the production of H2O2.)
The line you point out as a problem is a bit confusing, but it is entirely correct. It helps to keep in mind that 'glutathione' exists in two forms in the cell: glutathione disulfide (GSSG) and glutathione (GSH). GSH is the reduced form. So a reduction in GSH (the reduced form) does lead to an increase of H2O2.
Another way to say it is that in the absence of G6PD, the amount of reduced glutathione is decreased.
Glucose is used by RBC's as food, and G6PD is needed to metabolize it. G6PD reduces a molecule of NADP to NADPH + H+, which then reduces glutathione disulfide into GSH (reduced glutathione). That molecule is used in the reduction of H2O2 to H2O.
If there is a deficiency of G6PH, there will not be enough NADPH to reduce GSSG, leading to decreased levels of GSH; H2O2 levels rise, and oxidative damage to proteins in the RBC occurs.
Glutathione metabolism consists, in large part, of its oxidation to the disulfide and recycling to the reduced form by the co-enzyme nicotinamide adenine dinucleotide phosphate (NADPH) by mediation of glutathione reductase (Figure 1⇓). Reduced glutathione is important in the detoxification of free radicals. In normal cells, NADPH is regenerated by G6PD during oxidative stress. Impairment of this step prevents reduced glutathione recycling, exposing the cell to oxidative damage. Alternative pathways to G6PD-dependent NADPH production exist in most human cells but not in erythrocytes, and the lack of protein synthetic machinery deprives the erythrocyte of the opportunity to replace enzyme that has been lost. For these reasons, these cells are uniquely vulnerable to oxidative stress in G6PD deficiency.
These damaged proteins bind to the red cell membrane, which is then processed in the spleen; the spleen removes the damaged part of the RBC membrane. A few passes of this, and the RBC can no longer maintain the integrity of its cell membrane, and it breaks up (hemolysis).
Glucose-6-Phosphate Dehydrogenase Deficiency and Antimalarial Drug Development