CBT involves changing your cognitive patterns (C) and exposing yourself to your fears (B). The cognitive part is founded on the belief that an anxiety patient has what are called "cognitive distortions". The anxious person only sees the negative possible outcome or reason for a situation, and totally disregards the neutral or positive explanations for a situation. You work by recognizing your biased thoughts as it were and analyzing them and making a more balanced thought.
For example, a cognitive distortion might be "I embarrassed myself in front of Joe, and he must think im an idiot". Two cognitive distortions are easily recognizable here. First is "mind reading", you can't possibly know what "Joe" thinks because you can't read his mind. And second is your "labeling" yourself as an idiot, when in reality there is no such thing as an idiot.
Your more balanced thought instead would be something like "I may have felt like embarrassed myself in front of Joe but I can't possibly know what he thinks about me unless I speak with him. Also there is no such thing as a loser, so I can't be a loser"
The complete list of cognitive distortions are in the link.
Cognitive Distortions
The other part of CBT is the behavioural part (B) in which you (usually with the assistance of a therapist) expose yourself to frightening situations, in a controlled environment. It works on the fact that your anxiety cannot remain at 100% indefinately, and it MUST drop (even ever so slightly) eventually. You keep doing exposures till you either no longer fear them or your anxiety has dropped significantly. Usually one starts out with exposures that cause minimal fear/anxiety and work their way up their fear list to eventually their most feared situation.
In the end, CBT can make permanent improvements on your anxiety that might not be possible with medications alone.