The action of bits on the human pony, especially curb/leverage bits, is not always obvious.
You can think of the action of a snaffle bit as direct pressure. In the case of a snaffle bit, the reins are attached to the same bit ring that attaches the bit to the bridle itself, so when the reins are pulled, the bit is pulled directly back towards the rider/driver. There is no additional tension on the cheekpieces (the straps holding the bit to the bridle).
Unlike a snaffle bit, a curb bit uses leverage. The reins are attached to bit rings on the bottom of the mouthpiece, while the straps connecting the bit to the bridle (cheekpieces) are attached to the top rings.
The mouthpiece of any bit has a substantial impact on the severity or comfort of the bit. On a curb bit, in addition to the pressure put on the head and lips, the mouthpiece can put pressure on the tongue and roof of the mouth. As can be seen in the video at the top of this page, as the rein pressure is slowly increased, the mouthpiece rotates in the mouth.
For a bit with a port (as was used in the video above), this puts progressively more pressure on the roof of the pony's mouth as the port rotates from a position parallel with the tongue (essentially lying on the tongue), to, with enough rein pressure, a position perpendicular to the tongue. As you can imagine, the larger the port, the greater the pressure.
The rotation of the mouthpiece also puts pressure along the lips, and on the tongue if there is no "tongue relief"." Tongue relief means essentially, a space for the tongue to sit. In the video above, there is tongue relief, so it is not quite so severe. If, instead of an open-ended inverted "U" shape, the bottom of the U was a solid bar, there would be no place for the tongue to "escape" from the bit when the reins were pulled, and it would be a more severe bit.
In the video above, you can see the chin strap is not terribly tight, which allows the mouth to be opened. This is for illustration purposes only, so you can see the movement of the mouthpiece and port. In a training situation, the chin strap would be cinched down, so the pony could not open his mouth. In so doing, it makes the bit much more effective because it adds pressure to the chin, and it prevents a pony from opening his mouth to escape the pressure of the bit on the roof of his mouth (and on his tongue if the mouthpiece does not provide for tongue relief).
As I mentioned above, the differentiating factor for a curb bit from a snaffle is the use of leverage (versus direct pressure for a snaffle). What might not be obvious from the video at the top of the page is where the leverage comes from.
In the animation at the right, you can see the leverage is generated from the length of the shanks (the bottom "arms" of the bit - from the mouthpiece to the ring where the reins are attached). As the reins are pulled, the buttom of the bit (i.e. the end of the shanks) are pulled towards the rider/driver. Because the bit is attached the bridle via the top rings of the bit, the bit pivots about the mouthpiece, which itself rotates (illustrated in the video at the top of the page).
The result is that the distance between the top bit rings and the bridle decreases, thus increasing the pressure on the cheekpieces (which translates into pressure on the upper lips and the back of the pony's head). At the same time, the pressure on the sides of the pony's mouth also increases. If there is a port, pressure is also put on the roof of the pony's mouth, and even on the pony's tongue (if the bit does not offer tongue relief).
The length from the mouthpiece to the bottom ring (where the reins attach) is longer than from the mouthpiece to the top ring (where the bridle attaches). The lower arms of the bit are often called "shanks", and the longer they are, the greater the leverage of the bit (i.e. as they become longer, less rein pressure pressure is required to get the same effect).
As you can see in the illustration, the rein pressure gets translated into pressure onto multiple parts of the pon's mouth, lips, and head, maing for an extremely effective bit.