The song was written in 1963, and its meaning was first revealed in Newsweek in 1964.

Puff the Magic Dragon: puff on a cigarette; draggin' means inhale smoke

Jackie Paper: Puff's friend; what's a cigarette without paper?

Jackie brought sting & sealing wax & other fancy stuff: attempts to seal the cigarette paper if there is no glue available

Puff ceased his mighty roar: stopped smoking when there was no paper

green scales fell like rain: if there's no paper, the leaves get all over the place

slipped into his cave: used a pipe