What caused my toilet's flapper valve to become spongy and disintegrate (see pictures)?
Flapper valves in my bathrooms typically last about 2 1/2 years of so. Depending upon the type of material they either get mushy like the ones that you show or they start to get hard and no longer make a good seal. Since I do not add anything to the toilet water tank like a disinfectant or fragrant additive I have to attribute the degradation of the flappers to the water chemistry and the chlorine that the city adds to the water. 

Flapper valves are relatively inexpensive and the type you show in your pictures only take a few minutes to replace.