The standard way to think about anxiety probably goes back to the idea of "fight or flight." It means when we encounter a threat, we instinctually release hormones that prepare us to solve the problem by either fighting or running away. This phenomenon has been observed and empirically tested in many ways, and is also in line with common experience. I'd like to suggest an alternative, Torah way of understanding what it means, and perhaps where it comes from.
The running theory of where it comes from is that way back in the day, humans lived under conditions under which the fight or flight response was very useful. It was not uncommon to encounter mortal threats. Nowadays, our living conditions are much more advanced and controlled, and the fight or flight response is a vestige of our evolutionary heritage, but is of little use to us. In fact, it is harmful, seeing as prolonged stress can be deleterious for health and well-being.
In the Torah, the concept of life is much broader than the animalistic picture of survival of the fittest. For example, it says that the righteous are alive even in their deaths, and the wicked are dead even in their life-times. A person can be alive in a circumstantial way, merely in the sense that he happens to have a heart that's pumping. Or, he can be alive in the sense that he is always making himself into something which is flowing forward with vitality. When the talmud wants to say that both sides of an argument are true, it says אלו ואלו דברי אלוקים חיים (these and these are words of living G-d). To say that the words are an expression of life is a way of saying they are true because when something clicks and works with a bigger picture of reality it propels you forward and catalyzes you. A person that has this quality isn't alive because someone gave birth to him - he's alive because he is creating himself, and no one can take that away.
Everyone relates to this higher ideal of what "life" is on some level. That's why as we go through the vicissitudes of life we experience anxiety. We feel something doesn't click, something is taking away our ability to be alive. Depending on how we understand what being alive means, we may think about it in more physical or more spiritual terms and experience different things as threats. Very great people have reached hights where even literal threats to their physical lives were not as frightening as the possibility of turning away from the living G-d.
Unfortunately, we develop a very narrow picture of what it means to be vital, and we experience things as threats that really aren't. I'll explain more in the following posts, G-d willing, but in short, the process of overcoming anxiety has to do with addressing what seems like it doesn't click with our view of a meaningful big picture, and coming to see how it is really דברי אלוקים חיים.