Four ways and I can't tell which because your picture didn't come through....
1 a chrome plug(think chrome coated Jr Mint) on the bottom needs removed and you'll see a vertical bolts head.
2 there are small hex key holes on the bottom that release the cover(think toilet paper holder or coat hook)
3 the cover can be pryed off from the bottom(like the back of a phone.)
4 there's a bolt accessible from the backside of the panel it's mounted on. The bolt is centered between the four screws and has an access hole in the panel.
Swap the screws for through bolts with locking nuts.
If you don't want to remove the mirror:
There are screw nuts(like hold emblems and trim on older cars), generally stamped tin, which you can put on the existing showing screw and tighten.
A Nylon drywall anchor of the appropriate size will also work on the end of the existing screws.
If you use the drywall anchors,
Look up: how to apply safety wire on bolts
Then wrap the anchors accordingly. Just a hand wrap will do.