When I attended my first formal meditation program, only thing they asked us to observe was our breathing. First question which came to mind was why only breathing, why don’t they ask to observe our heartbeat or temperature sensation from our surrounding or digestion of food lying in our stomach.
There are many more natural processes inside human body which one can observe. The reason is simple, breathing is the only natural process which has both voluntary and involuntary components. You cannot change how much time your food will take to get digested or you cannot change your feeling of cold sensation if you are walking in snow. One natural process you can tinker with is breathing.
If you are feeling a sense of anxiety, your breathing rate increases but you can also learn to stay calm in such situation which will ultimately result in normal breathing.
Observing your breathing patterns initially provides you an opportunity to know how different thoughts and situations impact your breathing. In the process of recognizing impact of various thoughts on breathing, you learn to separate causes and effects and start taking more control on effects (i.e. breathing rate).
This changes everything
Now, since you have separated the causes and effects, you can shift from a state where feelings control the breathing to a state where breathing controls the feelings. This way you can program yourself to become calm no matter what is happening outside.