DIAGNOSIS The sound is a 60Hz hum from the internal transformer, transmitted through the wall. This is caused by a loose wire connection or a slightly undersized transformer in your HVAC system.
FIX Tighten all wire terminal screws on the wall plate. Remove each wire, inspect for fraying, and re-seat it firmly under the screw. Ensure no stray wire strands are touching adjacent terminals.
IF THAT DIDN'T WORK The HVAC system's transformer may be overloaded. This thermostat draws more continuous current than a basic model. You may need to install an isolation relay or a dedicated 24V transformer for the thermostat circuit.
HEADS UP A loud buzz or chatter indicates a relay sticking inside the thermostat. This is a defect. Contact Amazon support for a replacement under warranty.