The theory behind this is that by not eating before exercise, your body burns a higher percentage of fat. In a recent study at Northumbria University (Gonzalez et al, 2013), 12 athletes either fasted or had breakfast before a moderate intensity treadmill workout. The fasters burned 20% more fat than the breakfast eaters. Interestingly, throughout the rest of the day, the fasters ate less and did not have an increased appetite to compensate for lack of food pre exercise.
Another study (Proeyen et al, 2011) found that exercising on an empty stomach during endurance training caused metabolic adaptations to the muscles, so that they burned fat in preference to carbohydrate.
So, if the reason you are exercising is purely to lose weight, then an empty stomach may benefit you. However, in practise, if you find you lack energy/dizzy and are unable to exercise for as long or as hard, then the resulting decrease in calories burned may not be worth it.
As with many aspects of nutrition, there is no definite right or wrong, black or white. Find what works for you!
For those wanting to build muscle or improve athletic performance, then food intake pre exercise is vital to fuel the training session, and to facilitate the anabolic process of increasing muscle mass and increasing endurance (Chryssanthopoulos et al, 2002, Neufer et al,1987.)
Gonzalez J et al. (2013) Breakfast and exercise contingently affect postprandial metabolism and energy balance in physically active males. British Journal of Nutrition 23:1-12
Van Proeyen K et al. (2011) Beneficial metabolic adaptations due to endurance exercise training in the fasted state Journal of Applied Physiology 110 (1): 236–245.
Chryssanthopoulos, C. et al. (2002) The effect of a high carbohydrate meal on endurance running capacity International Journal of Sports Nutrition 12, 157–71.
Neufer P et al. (1987) Improvements in exercise performance: effects of carbohydrate feedings and diet. Journal of Applied Physiology, 62, 983–988.