The "kitchen" in pickleball refers to the 7 foot area from the net where hitting a volley is not allowed. Officially this are of the court is called the non-volley zone but players have all adopted the term "kitchen".
Just like in tennis, a volley is when a player returns a shot in mid air without letting it first hit the ground on his/her own side. In order to hit or return a shot from the non-volley zone, players must let the ball bounce on their side.
Allowing the ball to bounce on your side before hitting it while located inside the non-volley zone is difficult and is designed to add an element of challenge to the sport.