Depth of field is the range of distance in the photo. It controls what is in focus and how much is in focus.
A photographer should use depth of field because it gives them the option to have only one thing in focus or the whole photo
A photographer should use a small DOF when they want there subject in focus and the background blurred, commonly portraits.
A photographer should use a long DOF when they want everything in the background in focus as well.
Here are photos of the same subject at different apertures. We can see that the background goes from in focus to blurry as the DOF gets smaller. This makes the flag more in focus and stands out more than when you have a longer DOF.
Here I used a different aperture and then increased the distance between me and the subject. We can see the distance makes it harder to see the subject when the DOF is small but when you have a long DOF it includes more detail so it makes it easier to see the subject from farther away.
This is my favorite image with a small DOF because the ball is in focus but you can still tell the setting. I also like how the second ball leads you to the subject because it is significantly clearer.