how to export the black background photo to my hard drive, I can't do it?
Correct! and you never will. Lightroom (being a 'parametric' editor) CANNOT apply the appearance of the MASK / OVERLAY to the image.
The MASK is only a guide (a visual Overlay) to the area you are adjusting with the sliders.
Hide the Mask Overlay by unchecking the box to see the result of adjustments.
And change the color of the mask to something that is not misleading you (like low opacity green or red). Toggle the mask on/off with keyboard [O].
For your "Black background" -this is a job for Photoshop, where you can apply a Layer Mask (to select subject) that will allow you to show a layer (filled with 'Black' or any color) below the subject layer .