It comes from shuffleboard. In shuffleboard, the scoring zone at the far end of the board -- the one that can actually score points for your opponent if you land in it -- is called the "kitchen." When pickleball was invented in 1965 on Bainbridge Island, the creators borrowed that term for the non-volley zone near the net.
The logic carries over cleanly: the kitchen is a place you're not supposed to step into. In shuffleboard, it's because landing there hurts you. In pickleball, it's because volleying from that close to the net gives you an unfair advantage, so the rules say you can't.
There's also a folk explanation floating around -- the phrase "if you can't stand the heat, get out of the kitchen" -- and it fits so well that a lot of people assume that's the origin. But the shuffleboard connection is the more documented one.
So: borrowed terminology from an older court game, applied to a new zone with a similar "avoid this" function.