Dr. Stone was a real person and, indeed, the chief physician who attended to Lincoln and his family for medical needs while they lived in the White House. He was born in 1822 and died in 1872. His full name was Robert King Stone, and he also served as a professor of medicine at the 19th century Columbian College Medical School, now known as the George Washington University School of Medicine.
The term 'biggin' that Lincoln puzzles over for its meaning is, as Dr. Stone later reveals, a nightcap (i.e., a bed-time cap, not a drink). It's a word that William Shakespeare uses in his play to distinguish the onerousness of the King's crown from the more pleasant aspects of wearing a simple cap on the head for sleeping.
The passage Lincoln reads to Dr. Stone comes from William Shakespeare, specifically Henry IV - Part 2, Act 4, Scene 5. It is a soliloquy that Prince Henry renders while watching his father, King Henry IV, sleep. Prince Henry is eyeing the crown that lies beside his father and contemplates the significance of wearing it.
The actor Lincoln so glowingly mentions to Dr. Stone is Junius Brutus Booth, a famous Shakespearean actor and, as fate would have it, the father of John Wilkes Booth, also an actor and infamous assassin who shot and killed President William Shakespeare in Ford's Theatre in Washington, D.C., on April 14, 1865.
This was the first time the director introduced the episode instead of merely appearing in the opening titles.