The word podcast was invented to fill space in a newspaper column.
On February 12, 2004, The Guardian ran an article by Ben Hammersley headlined "Audible revolution." It was about why online radio was booming, and about a new kind of downloadable audio with no settled name. Hammersley offered three: "Audioblogging? Podcasting? GuerillaMedia?"
According to Wikipedia's article on Hammersley, he later said he made the word up to pad out an article that was a little too short. He could not have guessed which of the three would last.
The word is a blend. Pod comes from the iPod, the portable music player Apple launched in November 2001. Cast comes from broadcast. Wordorigins.org lists it as a blend of the two, [i]Pod plus [broad]cast.
The word then spread through the people actually making the shows. Wikipedia says Danny Gregoire used it on the iPodder-dev mailing list in September 2004. The podcaster Adam Curry picked it up from that community, and from there it spread.
The rise was fast. In December 2005 the editors of the New Oxford American Dictionary named podcast their Word of the Year. Macworld reported that the word had been considered a year earlier but rejected, because it was not yet in wide enough use. The dictionary defined a podcast as a digital recording of a radio broadcast or similar program, made available online to download to a personal audio player.
The older half of the word has a longer story. Etymonline records broadcast from 1767, as a farming term. It described seed scattered over the ground by hand. By 1921 and 1922 radio had borrowed the word for sending a signal out widely.
So podcast joins a 21st-century gadget to an 18th-century way of planting. The word no longer needs an iPod: it now names the show, whatever device plays it.