Carl Elliott in The Atlantic:
Beta blockers seem to be especially good performance enhancers when the performance in question involves an anxiety-producing public setting. This is because a large part of the anxiety of performing in public comes from the worry that one’s anxiety will become outwardly obvious. Most people who worry about public speaking, for example, aren’t worried that they’ll flub their lines, trip and fall as they approach the podium, or deliver an hour-long speech on television with their pants unzipped. They worry that their anxiety will become apparent to the audience. They’re terrified that their hands will tremble, that their voices will become high-pitched and quivering, and that beads of sweat will appear on their foreheads and upper lip, like Richard Nixon trying to explain Watergate. This is why beta blockers are so useful; people who have taken a drug that blocks the outward effects of their anxiety become less anxious—not because the drug is affecting their brain, but because their worst fears are not being realized.
Beta blockers have been around since the 1960s, but it took a while before anyone noticed how useful they were for performance anxiety. Probably the first performers to start using them widely were musicians, especially classical musicians, whose hands can get clammy or tremble during a concert performance.