This is about Doctor Robert Freymann, a "Speed Doctor" in New York who supplied many celebrities, including The Beatles, with drugs. Reportedly, Dr. Freymann, who died in 1987, treated patients with injects of amphetamine-spiked vitamin B-12 from his Manhattan East 78th Street clinic. It should be noted that most Americans, due to diet and convenience culture, suffer from a vitamin B deficiency, and so an infusion of B vitamins feels like a "pick me up" all by itself, which is why many popular "energy drinks" (Red Bull, Monster, etc.) have this in the ingredients.
The Beatles recorded this on Sunday, April 17, 1966. Two days later, they chose Take 7 to overdub all the vocals. (thanks, Ben - Cheverly, MD)
John Lennon did indeed peg this song as autobiographical, stating in interviews that he was the one who carried a pocket full of pills on tour. However, both he and Paul McCartney deny going to this specific doctor, but say that this song was kind of a piss-take at the idea, which struck them as funny.
John's vocals are double-tracked and then split into two slightly out-of-phase stereo channels to simulate a trip on drugs. Also, the pivot to the key of B during the bridge gives a sudden blissful shot to the song.