You missed a dot. v dot is the rate of change of lateral velocity, and he's assumed V is constant. v~V*beta, so v dot= V*beta dot
In English rate of change of lateral velocity equals vehicle speed times rate of change of vehicle slip angle.
As you might imagine it's a fairly small proportion of ay in typical scenarios, unless you are drifting or losing the back end. beta dot is a noisy measurement in reality.
The final line ay=V*(r+beta dot) boils down to lateral acceleration= vehicle velocity*(yaw rate+rate of change of vehicle slip angle). So if you haven't got a Corevit then one way to get beta would be to integrate betadot, (ay/V)-r all of which you can measure with traditional instrumentation. Beta is handy, it tells you what your rear tires are up to.