DIAGNOSIS That's the sound of the burr grinder's plastic gear train stripping out. The primary drive gear, which is made of nylon, is a known weak point and wears down or cracks, especially if you've run it with rocks or very hard debris in the hopper.
FIX You'll need to replace the grinder assembly. The part is officially called the "Grinder Housing Assembly" (Cuisinart part number SP-357190 for many DGB models). It's a single unit you swap in. To access it, remove the bean hopper by twisting it counter-clockwise and lifting off. You'll see several screws holding the top housing. Remove those, lift the top, and you'll see the grinder unit held by a few more screws and a wiring connector. Unplug, unscrew, and replace with the new unit.
IF THAT DIDN'T WORK If a new grinder assembly still screeches, the motor shaft itself could be bent. At that point, it's usually more cost-effective to just use pre-ground coffee with the machine (select the 'OFF' setting for the grinder).
HEADS UP When you get the new grinder assembly in, never run it empty. Always have beans in the hopper. Running it dry accelerates wear on those nylon gears.