What does “dragging the shutter” mean when using flash?

The idea is that you have two sources of illumination: the ambient light and the flash.

Because the flash is such a short duration, if you change your shutter speed from say 1/200th to 1/100th of a second, it will not affect the amount of flash exposure since that is a constant amount of light that will occur in a fraction of those exposures.

However changing the shutter speed (keeping aperture constant) will affect the amount of ambient light. So as you lower the shutter speed, you will lighten the overall exposure, while anything that is mainly illuminated by the flash will remain relatively constant.

The second concept is that the flash will only illuminate things close to the flash, but will not affect the background, assuming the background is relatively far from the camera.

So when you put all this together, by changing the shutter speed you can affect your background (as if the flash wasn't there, since the flash won't affect your background) while your subject will have about the same exposure.

So then dragging the shutter is a way of having a well lit subject, mainly due to flash, but then being able to artistically adjust the background. If there is a sunset you might want to darken the background to give a more dramatic, saturated look for example.

If your experiments at home failed, this could be due to a few reasons. First, your subject needs to be relatively close compared to your background. Otherwise the flash will light up the background and the ambient light will be minimized and you won't see the effect. Also, as you change the shutter speed, you'll want to keep aperture fixed. So I'd use Manual mode, otherwise the camera will adjust aperture and the ambient exposure will remain constant, which isn't what you want if you are dragging the shutter.