How to remove the wall plate behind Honeywell thermostat
The back plate was meant to be held on by screws into the plastic anchors visible on one of the photos.

If the screws are out and the plate is still attached to the wall, it is likely that paint or compound are holding it on.

Try scoring around the edges with a utility knife, then slipping a putty/taping knife behind the plate and prying it off. There may be some minor cosmetic damage to the drywall or paint.