There is a common misconception that this muscle is a supinator of your forearm – hence its “old name” supinator longus -.
The brachioradialis brings your forearm back in a neutral position after it has been supinated or pronated. Thus it acts like a neutralizer.
EMG studies showed that it only supports the supination and pronation out of a neutral position if there is additionally external resistance. This is the case when you tighten or loosen up a screw with a screwdriver, for example.
Furthermore it stabilizes your wrist when you are gripping something hard. This stabilization work prevents your wrist from bending, which is the motion that the flexors of your hand and wrist would initiate with a gripping motion.
Beside its stabilizing function of the wrist, this muscle also stabilizes your elbow when you are moving it very fast and high centrifugal forces develop. This might be the case when you perform a hook punch in boxing.