People with anxiety commonly suffer from excessive sweating. There are two main reasons that this happens. The first is actually a consequence of a another anxiety symptom, hot flashes aka heat flushes. Go to the hot flashes page to read more about that. The second is a more general phenomena associate with anxiety, which is activation of the sympathetic nervous system.
The peripheral nervous system (everything outside of your brain and spine) is divided into two opposing branches. Literally you have two nervous systems outside of your brain and spine. People with anxiety disorders tend to have a chronic over-activation (meaning the ‘resting’ level is quite high) of the sympathetic nervous system.
The sympathetic nervous system is basically the body’s GO mode. It is associated with high levels of activity, in contrast with the parasympathetic nervous system. When you are feeling anxious, your sympathetic nervous system becomes activated, it’s an essential part of feeling anxious.
When the sympathetic nervous system is activated, there are a large variety of physiological consequences throughout the body. There is typically vasoconstriction, particularly around your organs. The body moves blood from the core part of your body for the vital functions, and redirects the support to your limbs. Another aspect is sweating! This is why you sweat, it is a sign that the sympathetic nervous system is being activated.
When psychologists or cognitive scientists are measure activation of the sympathetic nervous system, they use a measure called Galvanic Skin Response (GSR). It is also known as Electro Dermal Activity. This measure is taken by placing electrical leads on the body, and passing a very low current between them. Based on the amount of sweating your are doing, the electrical resistance in your skin will rise and fall.
The more anxious you are, the more sympathetic activation you have, and the greater your skin conductance. In psychopathology studies that look at the anxiety disorders, GSR is considered an index of sympathetic activity, and thus how anxious you are.
To recap: just like any mental state, there are consequences in the body. As anxiety and worry go up, the sympathetic nervous system is activated, as it is one of the primary responses your body has to stress. This is why you start sweating when you start feeling anxious. It is not specifically associated with the anxiety disorders, but obviously is a key trait for anyone with an anxiety disorder.