With a certain distance to the subject, it is in focus at a certain distance to the image sensor. As you move closer to the lens, the place it is in focus moves back. this is because the lens property to bend light is in principle fixed and as you move closer, you change the incoming angles. Naturally, something has to compensate for this, and the distance form lens to sensor plane is one way. And the beauty of this solution is that it is easy for you to do.

When you focus your lens, it will also move the lenses. Old and simple designs move all the lenses away from the sensor as you focus from max to minimum distance. See [left] far focus [right] close focus:

You can see the entire lens holding group moved further into the lens house.

They have to make a choice to stop the mechanical movement somewhere. But when you add a spacer ring to move the entire lens further away, you can focus closer, at the expense of infinity - because now the lens cannot move close enough.

New internal focus moves exploit that the "objective" is made form many "lenses", and it displace them relationally to keep the focus on the sensor distance.

You can say that the oldschool method acknowledge and makes use of the fact that closer subjects are on focus farther away from the lens, and the internal focus change the angle bending properties to compensate.

You might then ask, why they often stop moving the lenses at around 50-85mm minimum focus distance. Are they sneaky and want to sell you expensive macro lenses? Obviously, size is a competing factor. Also, when moving to macro work there are some optical challenges that needs to be fixed other than the min. focus distance such as flatfield focus. It this range the DOF is extremely narrow and for macro you want corner sharpness. You also want a longer manual focus span for finetuning of where to place that narrow plane.