Chambering is an analogy drawn from firearms. In firearms, you chamber a round (put a bullet in the chamber). In the same way as you chamber a round in firearms, you can chamber a kick to "fire" it. This is very probably an American saying in martial arts.
When one refers to chambering a technique (any technique, but in this case a kick), they mean fully drawing back the technique in preparation for exploding forward. In looking for images to illustrate this point, I was disappointed because I cannot find an example of a good chamber for a sidekick. So I'm going to ask you to use your imagination.

In image #4 this is what we mean by chambering a sidekick. This stylist is illustrating a poor chamber. His knee should be tightly tucked into his chest instead of far out from his body, and his heel should be pointing directly out at his target. Thus, when executed, the sidekick should pump quickly and powerfully out, then retract back in like a piston.
EDIT
To further illustrate the point, the following is a sequence of throwing a front kick. There are twelve images. If we number those images top to bottom, left to right, the 9th and 11th images are chambers for a front kick.

Likewise, in the image series below, images #10 and #12 are chambers for a round kick.

A chamber is, very simply, the inverse of the executed technique. To use a punch as an analogy, the fist on the hip is the chamber, whereas the fist that is punching is the technique. Notice that the chambered fist is exactly the opposite of the punch. It is supinated, whereas the punch is pronated. When the chambered fist is thrown it will twist. There is some yin/yang stuff going on here, but it need not be thought of esoterically, but rather as physics.
Chambering adds power to the kick
By chambering the kick fully (especially in the example of the sidekick) you are adding power to your kick in two ways:
- By fully drawing back, you are increasing the distance your kick will have to travel. The further it has to travel, the more speed it will develop en route to its target, and therefore the harder the impact will be. This is a game of inches, so drawing back even a little more than you could before will help.
- By fully drawing back, you are targeting the area of your opponent's body that you want to hit. Your kick will follow a straight trajectory to the target. Any deviance from a straight trajectory (i.e. "sweeping your kick") will detract power from your kick. This is true in round kicks and other circular kicks as well, though in those cases the trajectory is circular, not straight, but still should be as direct as possible and deviate as little as possible from its trajectory.