There are a number of factors that contribute to stress... Mental, Physical, Emotional (and perhaps Spiritual).
As mentioned above, exercise can increase endorphins and 'burn up' or convert chemicals and hormones that are part of the chemical cascade of Stress. These 'Stress Chemicals' are designed to increase our ability to survive, primarily through movement (think: run, fight).
Our current lifestyle generally does not often include much physical exertion but it does includes a lot of stimulation that triggers the Stress Response. Therefore we are building up the 'I need to Move to Survive Hormones and Chemicals' but not moving much to utilize them properly.
Movement does more than 'burn up' stress hormones... When we walk we stimulate a 'cross crawl' patterning in our brain which tends to balance the electrical activity which helps change our perception and therefore help us feel better. It is possible with brain wave training to 'unstress' in a few minutes. An accomplished meditator can do this.
When we are stressed our attention is on problems (past, future) and therefore feel the emotions and feelings of worry, fear, anxiety, grief, anger and frustration.
When we exercise there is a good possibility that our attention is on the present moment which allows feelings like joy, appreciation, happiness and being 'connected to something greater than ourselves' to be present. This in itself changes the chemicals that our bodies make which then affects how we feel.
In Chinese Medicine there is a lot of understanding and practice of balancing the 'Chi' or energy in the body. When we move we change the blood flow and the 'Chi' flow which directly affects how we feel. An acupuncture treatment can often quickly affect how you feel without the movement of exercise.
Part of the stress response is a contraction of various muscles... unconscious and instinctive. When you exercise often it will disrupt the unconscious contractions you are doing and then you will feel more relaxed, unburdened and more free.
On another level, it is possible to measure various kinds of electrical activity of the body (like brain, heart, muscles) using various BioFeedback techniques. Exercise changes these electrical patterns, usually for the better!