Impatient, edgy, fidgety, huffy and half desperate. One who is chafing at the bit wants to GO, and RIGHT NOW. They stand in the doorway of your room and stare at you as you tie your shoes, shifting their weight and exhaling in sudden loud bursts. They look at the clock and make pointed references to how late it's getting, meaning how late you are making them (even if you aren't late in the slightest). Then they give up and go stand on the porch until you come out the door. By this point they may be smoking out of sheer irritation.
This term seems to originate in equestrian circles. The bit is the part of a horse's bridle which goes between the teeth, providing the rider with control over the head. The rider will use the reins, which are attached to the bit, in order to direct the horse. But if a horse does not like its directions--especially when those directions are to hold still in the box before a race--it may express itself by straining against the bit. This may cause chafing in the mouth area, especially at the corners, where the bit actually hits skin (thus the physical area is called "at the bit"). In short, they chafe themselves, causing themselves extra pain through their own frustration.
So, by chafing at the bit, the impatient one just increases their own irritation. Many chafees are aware of this. This is not going to stop them, however. As inconvenient as it might be, you cannot just smother impatience. Some people deal with it better than others, though. I personally deal with it very badly indeed.
The phrase "champing at the bit" is also used, but seems to me to imply less irritation and more simple eagerness. There is no pain inherent in champing.
I am not a horse person--if anyone has a more accurate explanation of the function of the bit, please /msg me. Thanks.