This myth is probably 2-fold. First, if you have food in your stomach then the body will be too busy digesting that it will not be working. Second, if you have recently eaten you won't be burning fat. Both of these thoughts can be addressed by one process. When we eat something, the body sends a lot of resources to help digestion.
Digestion is considered a stressor so the body tries to resolve it relatively quickly.
So, the body's resources are sent to deal with the exercise stress first and digestion is essentially put on hold until the exercise is completely. This seems to be a really popular myth especially at health clubs. This was really popular about 10 years ago with the media attention on preventing heart attacks. It became almost a rule and standard. Everyone can find a block of 10 minutes in his or her day to exercise.