> S301 orbits Sgr A* so tightly that the star’s trajectory should be bent by the black hole’s rotation.

Simple version: They'll have to measure S301's orbit with all the precision they can, then solve a bunch of tricky equations to figure out Sgr A*'s spin from that. Like all measurements of distant orbits, there'll be a lot of uncertainty in their observations - so it's a game of measure, calculate, wait, measure again, calculate a bit-less-uncertain number, wait, repeat.

Hard version: The physical mechanism for Sgr A*'s spin changing S301's orbit is explained by Einstein's Theory of General Relativity. There are plenty of hand-wavy explanations for laymen out there...but the Real Deal is Ph.D.-level physics.