When we talk about following your subject, we don't mean stalk them (unless you aspire to joining the paparazzi!).
What we mean is, whatever you are trying to capture, stay with it until the end. If it is an animal running or a person surfing, a race car on a circuit, or even a bird flying, "keep on it" until it has exhausted any photo opportunities or gone out of sight. Follow the subject with your lens, trying to keep it a smooth flowing movement, rather than a series of short, sharp, move-stop-move motion
The previous subject "Use both eyes open" is a definite help here as you want to be aware of what your subject is doing at all times. The more you see, the closer you are to getting the best shots that you can.
It is all too easy to flit and switch your vision to something that could be more interesting only to find that monkey that you were looking at pulls the cheekiest of grins, or does something well worth capturing, when it is too late! Follow the subject and stay with it!
Try to keep your shutter finger half pressed so that the camera is constantly altering the exposure and focus as you move the camera around. Many D/SLR have various focusing modes, (Single/ Manual/Continuous, or similarly named). Read your manual on setting these up on your camera. You will find that continuous focusing is good for moving subjects.
Now, if I am out and about with the purpose of taking pictures, the camera is always in my hand and constantly switched on. With practice, you become more aware of your surroundings and feel ready to pounce when the action strikes. Here are some examples of how and why to follow the subject.
You have visited a beach and found several surfers. It would have been all to easy to jump from one to the other but by following one particular surfer until he had finished his run, you will walk away with more 'keepers' than if you just snapped away, jumping from one surfer to the next. Even at the end of a surfers run, you can get some great candid shots as they "exit" their boards in sometimes spectacular fashion.
Hopefully this will give you inspiration and motivate you to really study your subject and walk away with a lot more "keepers". So learn to follow your subject, keep both eyes open
and you will soon be getting some great action shots.