How does MSMA kill crabgrass with only minimal damage to lawn grasses?
Please consider this a sort of comment — I tried looking up for articles related to MSMA mode of action but almost all the articles are about its toxicity in mammals.

By what I understand, trivalent arsenates act as phosphate analogs and therefore interfere in phosphorylation reactions including DNA synthesis [Ref]. 

The weed is affected perhaps because of its relatively higher growth rate. Also the already established lawn grasses would have good population size to regrow out of the damage.