If you picture that, you'll be picturing a rectangular box. The tee box. But that's not actually how "tee box" came into usage. Golfers began referring to the teeing ground as a "tee box" because - in the days before wooden golf tees - the most common method of teeing a ball was on top of a small mound of wet sand. The sand was made available to golfers inside boxes placed on each teeing ground. And what's a box that contains sand used to tee the ball? A tee box. (For more on this topic, see our history of golf tees.)
The term "teeing ground" refers to one, specific set of tees. "Tee box" is also used to refer to one, specific set of tees, but is also used to refer to the full complement of teeing grounds on any given hole. A golf course might have three, four, five or more sets of tees of varying yardages. Often, several of those teeing grounds are grouped together, and "tee box" can refer to that grouping as well.
