Etymology of ~Getting into someone's “kitchen”~

Solution 1:

The Historical Dictionary of American Slang quotes the 1989 edition of Paul Dickson's Baseball Dictionary for the definition of kitchen:

The area of a batter's torso inside or at edge of the high and inside portion of the strike zone....

This is the target area for a brushback pitch, one intended to force batters away from the plate. The pitch is not intended to hit the batter but to be close enough to force batters away from space they've claimed as their own, space metaphorically like their own kitchen. The HDAS goes on to state

The term is used in more elaborate metaphors: "He got in his kitchen and broke a few dishes."

A brief search in references and the google does not reveal whether this is the origin of the "trash talk."