They used to be used for TV transmitters, too, though most are now solid state. A while back, the San Jose PBS station, KTEH, had a disaster when their klystron tube froze. They had a hell of a time even finding a replacement tube, and had to beg extra-hard to pay for the damn thing. Somewhat weird that the PBS station in the Silicon Valley would be using an antiquated transmitter. They’ve since had a drive for a new transmitter.
Basically, they’re for generation of high-frequency radio waves. The following is a blurb about how they work:
One of the resident physicists may be along to explain their use in particle accelerators and so on.
And, just to eradicate your ignorance about Useless Trivia, the etymological derivation of the word “klystron” (courtesy of the American Heritage Dictionary):
*A “clyster”, by the way, is a now-rare medical term for an enema.