Performance anxiety, or stage fright, develops from how you think about performing in front of an audience. It is not the stage and performance that causes you anxiety- rather, it is your thoughts about the performance situation which leads to anxiety. So your irrational thoughts about the stage performance may include: "They will make fun of me;" or "I will embarrass myself;" or "They will not like me."
So these thoughts then induce anxiety, which also includes the physical symptoms of anxiety from the release of adrenaline into the bloodstream. The physical symptoms of anxiety include racing heart beat, pounding chest, increased rate of breathing, shortness of breath, sweating, trembling, lump in the throat, and butterflies in the stomach.
But these anxiety symptoms are distressing, so your natural reaction is to avoid the performance situation. But the avoidance only serves to maintain the belief that people are making fun of you, and you never get to find out that if you just get up on stage and perform, the anxiety will just go away naturally, on its own. So the avoidance maintains this negative, vicious cycle of performance anxiety.
So performance anxiety comes from the way you think about performances, and how you avoid those performance situations. So to break this negative cycle of performance anxiety, you have to change the way you think, and you can't avoid the feared situation.