Salts with asymmetric (fluorosulfonyl)(trifluoromethanesulfonyl)imide (FTFSI) anions have recently been shown to suppress crystallization of water-in-salt electrolytes, enabling low-temperature operation of high-voltage aqueous rechargeable batteries. To clarify the underlying mechanism for the kinetic suppression of crystallization, we investigate the local solution structures and dynamic behaviors of water-in-salt electrolytes based on the asymmetric FTFSI anion and its symmetric anion analogues by Raman spectroscopy and molecular dynamics simulations. We find that monodentate coordination of FTFSI to cations leads to high rotational mobility of the uncoordinated SO2CF3 group. We conclude that the peculiar, coordination-dependent, local dynamics in the asymmetric FTFSI anion, manifested by enhanced intramolecular bond rotation, enables the strong supercooling behavior.