1) It's always easiest to fold the Brompton if you are standing on the left side of it. The first step involves parking the bike by putting it into its kickstand position. To do so, turn the front wheel slightly to the left (thereby getting it out of the way and protecting its fender) and release the catch for the rear wheel by pulling the lever under the seat. Raise the bike by the saddle, allowing the rear wheel assembly to swing down and forward, ultimately lying under the main frame. The bike is now in its kickstand position, and providing that it is on reasonably level ground will stand up unsupported.
It is tempting to fold the seat or handlebars next. However, the bike cannot be folded properly in this order: you must fold the front wheel assembly back first.
2) When folding the front wheel back, make sure that the righthand pedal is out of the way. It is best if the righthand pedal is pointing backwards and downwards. If you are viewing the bike from the right side, it would roughly be in the 7:00 or 8:00 position.
3a) Unscrew the hinge clamp on the main frame until the locking plate is free (three or four turns of the knob normally does the trick). Grasp the handlebar with your left hand, lift slightly (pushing down on the back of the saddle at the same time with your right hand) and, keeping the front wheel pointing forward, use a "stirring" action to move it out away from you and then back around towards you in an arc until the front wheel lies alongside the rear wheel. If it's easier, just lean in and pick up the front wheel by hand and guide it back into place.
3b)As you've noticed, there is a plastic hook on the left side of the front wheel. When swinging the front wheel back and aligning it with the rear wheel, make sure that the hook has passed over the stay on the rear frame, and allow the front wheel assembly to lower into place with the hook resting on the frame, securing it in place. The front wheel is now in its folded position, and you can proceed with folding the rest of the bike.
4) Next comes the handlebars. Loosen the hinge clamp at the base of the handlebar stem; once loose, swing the handlebars down to lie alongside the front wheel and push them in so that the plastic nipple engages in the catch, securing them in place.
5) The saddle is next. Open the quick-release lever at the base of the seat post and slide the seat post and saddle down. Make sure to slide it down as far as it will go, as this will lock the folded bike in place. Leaving the seat post partially extended can result in the bike remaining unlocked and then having it partially unfold.
If you are using EZ Wheels, the bottom of the seat post has a rubber stopper to prevent it from rolling away. To roll the bike, simply lift the seat post up about half an inch.
6) Finally, you have the folding pedal. Simply reach underneath the lefthand folding pedal and lift the black plastic center piece up. As you press up, the pedal will clear the crank and fold into place. Simple enough!
Here are two videos featuring the Brompton fold: