Let's take an example.
Suppose you have the camera and flash, and then you have your subject at 10 feet away. Then the dark gray backdrop is 12 feet away. There isn't that much distance difference between the subject and the backdrop, so the amount of light reaching each is similar. As a result, the backdrop will be fairly visible, even if it is dark in tone.
Now, suppose you have your subject at 6 feet and the backdrop at 12 feet. There is much more difference, so the light getting to the backdrop is much less than the subject. The camera's metering will set for the subject more than the backdrop, and the result is the backdrop will appear much dimmer.
All that assumes that there is a flash on your camera. Now, suppose you change to two flashes. Move one a couple of feet to the left of the camera, and move the other one to a couple of feet to the right. If you aim them correctly, they will both illuminate the subject and not that much will scatter all the way back to the backdrop. You can fool around with that and get the necessary result.
---Bob Gross---