At stall (when the rotor of the motor is not spinning), brushed DC motors generally behave like a resistive load (so the stall current is proportional to the supply voltage). Motor drivers like the VNH5019 can disconnect the motor from the supply to decrease the current from the stall current. Current is not pushed through the motor by the driver; the driver just supplies as much current as the motor draws. Increasing the maximum current the motor draws would require increasing the supply voltage. Generally, motor drivers do not do that.
The sentence you quoted means that if you have a motor that tries to draw more than 12A for too long, the VNH5019 might overheat (this driver turns itself off when that occurs). If you have a motor that tries to draw more than 30A, the VNH5019 will detect that and quickly turn itself off briefly to limit the current.
In general, it seems like the VNH5019 can handle enough current to work with your motor (we generally recommend using a driver that can handle the stall current of the motor continuously). Also, you should be aware that a mechanical load that stalls the rotor (prevents it from spinning) can cause the motor to draw the stall current, that drawing the stall current can damage the motor, and that the VNH5019 will not prevent the motor you mentioned from drawing the stall current in that situation. Generally, you should avoid stalling the rotor to prevent that.