I've always believed that most people experience stage fright to some extent when speaking in public. Some manage it well and may even feed on it, while others suffer sweaty palms and racing hearts.
This video by Mikael Cho explains what stage fright is and suggests that if you understand it you are more likely to overcome it.
It's interesting and beautifully animated. Source: Ted-Ed - The science of stage fright (and how to overcome it).