The Doctor once said, "scared is a superpower." In other words, being afraid can make you a better, more efficient human. And it can (Note: If you don't know who I mean when I say "The Doctor," know that there is only one, and you should really get to know him).
Ultimately, fear is a fuel for your body. When you are frightened, the hypothalamus sends your body signals to start producing adrenaline (and other stress hormones that are less well-known). These hormones increase heart rate and respiration, which prepares you for flight (if needed) and ensures that your body is oxygenated. In addition to this, the production of insulin is impeded, and as a result, there in an increase in the synthesis of sugar and fat, which the body can use as additional fuel.
No, you won't be able to fly or become invisible just because you are scared; however, your body is better prepared to move and to act. And of course, this can be a very bad thing for those who have significant heart conditions, as the increased stress can lead to a heart attack. But as a general rule, fear is a tool for survival.