The problems DOE addresses are nearly universal. Perhaps the best way is to look at a problem to which DOE does not apply and show why it doesn't apply. When the Levy-Shoemaker comet crashed into Jupiter, it answered a lot of questions. Astronomers could carefully observe and record the results of each impact, but they had no control over the size of the pieces, the angle of impact, over anything in fact. In addition, the impact of the first piece made changed the Jovian atmosphere enough so that the second impact was an entirely different kind of event.
Astronomers have no control over the events that they study.
An astronomer could use DOE with simulators, of course, but in the real world, the lack of control is a critical obstacle. A good experiment design gets you the answer you need at a price that you can afford to pay. A better experiment design gets you the answer you want for less than you were willing to pay. The best experiment design gets you the best answer at the least possible cost. Let's start with the elements of a good design.