DIAGNOSIS That's the sound of the internal thermostat cycling on and off to regulate temperature. A low hum or occasional click is normal. A loud, continuous buzz or hum, however, can indicate the thermostat contacts are arcing or sticking slightly.
FIX Unplug the grill and let it cool. Ensure the grill plates are seated correctly and that nothing (like a bit of foil or food debris) is stuck between the plates and the heating element, causing a vibration. Plug it back in. If the loud buzz returns only during the initial heat-up and then stops when it reaches temperature, it's likely the thermostat on its way out but still functioning.
IF THAT DIDN'T WORK If the buzzing is very loud, persistent, and accompanied by a burning smell or flickering lights, the thermostat is likely failing and causing electrical arcing. Unplug it immediately and do not use it. The unit needs a thermostat replacement or should be retired.
HEADS UP Don't ignore a new, loud electrical buzzing sound from any appliance. While it might run for a while, it's a sign of internal electrical stress that could lead to a complete failure or, in rare cases, become a fire hazard.