Depending on the model, they either use an engine driven pump thru a control valve to control torque to the wheel drive motors. Or like some of the Toro's I was looking at they belt drive from the engine out to hydrostatic drive units at each drive wheel. These individual hydrostatic units are similar to the drive units on a lawn/garden tractor. They contain a pump and motor and the hydraulic/lube fluid. I think most of those hydrostatic units are variable displacement pumps and vary the ammount of fluid they pump and which direction it goes to the motor based on the command lever position. This makes for much finer/smoother control.
As mentioned, they most likley do not have brakes, and if they do, it is probably an internal wet type brake inside the hydro unit.