Why is potassium permanganate used to kill mollusks?
The provided link (Lab Tutorial at Washington University in St. Louis)  states the following:

Another problem for water-treatment plants in the Great Lakes and Mississippi Valley is zebra mussels, which can &quot;congregate&quot; at the water intake and clog the screen, eventually sealing it off. Potassium permanganate, KMnO4, can be used to kill these mollusks.

Zebra mussels are a problematic invasive species in the midwestern USA. Permanganate has been found to inhibit their proliferation. The main effect is apparently oxidation. From a report released by the US military (ref [1.]) on use of oxidizing molluscicides including permanganates:

In general they have similar modes of action based on the oxidation of organic matter,which leads to toxic and lethal effects. They are suitable for use in preventative treatment, where they are added to a system throughout the breeding season at from 0.1 to 0.5 mg/L (ppm) total residual oxidant (TRO) to prevent settling. For reactive treatments,continuous application of 0.5 to 1.0 mg/L TRO for 2 to 4 weeks can eliminate established adult colonies, but concentration and contact time required depends on temperature, water chemistry, and physiological state of the zebra mussels. Mussels do detect oxidants, and shell closure for up to 2 weeks may reduce efficacy in adults(Claudi and Mackie1994)

See also ref [2] cited in ref [1.].
Ref [3] compares the toxicity of various molluscicides, noting that

An  LC50 value  could  not  be  estimated  for antimycin,  Bulab  6002,  Bultab  6009.  Calgon DM -DACC,  KML  V2.  KML  V54.  potassium  chloride, and  potassium  permanganate  due  to  insufficient mortality  of  the  test  organisms.

In addition, potassium can also have an effect on mollusks. For instance, from a thesis (Ref [4]):

Potassium can reach increased levels that are toxic to aquatic organisms, such as mollusks, in the environment (Romano and Zeng 2007). For example, Fisher et al. (1991) reported that increased potassium concentrations were toxic to Dreissena polymorpha, causing asphyxiation by damaging gill epithelium. In addition, Imlay (1973) suggested that high potassium concentrations could be the cause of absent mussel populations or lower diversity within Upper Midwest mussel assemblages. Any river or stream with a potassium concentration of 7 mg/L or higher lacked mussels, but mussels could be found in rivers with a concentration lower than 4 mg/L.

Note responses vary across organisms. Discussion of that is bound to take us increasingly away from chemistry.
References

Zebra mussel chemical control guide (ERDC/ELTR-00-1) by Susan L Sprecher, Kurt D. Getsinger; prepared for U.S. Army Corps of Engineers (2000).

Klerks,P.L., Fraleigh, P.C,and Stevenson, R.C.(1993).&quot;Controlling zebra mussel {Dreissena polymorpha) veligers with three oxidizing chemicals: Chlorine, permanganate, and peroxide+iron.&quot;Zebra mussels: Biology, impacts,  and control. T.F.Nalepa and D.W.Schloesser, eds., Lewis Publishers, Boca Raton,FL,621-641.

Diane L. Waller, Jeffrey J. Rach, W. Gregory Cope, and Leif L. Marking, Susan W. Fisher, Henrycka Dabrowska. Toxicity of Candidate Molluscicides to Zebra Mussels (Dreissena polymorpha) and Selected Nontarget Organisms. J.  Great  Lakes  Res.  19(4):695-702 Internat.  Assoc.  Great  Lakes  Res.,  1993.

Gibson, Kesley. (2015). ACUTE TOXICITY TESTING ON FRESHWATER MUSSELS (BIVALVIA: UNIONIDAE) AND FRESHWATER SNAILS (GASTROPODA: CAENOGASTROPODA).