Why is the freezer making a very loud buzzing noise after being unplugged for a few days

We had a similar odd sound from the back of our refrigerator.

There's a few different parts back there that can make noise: The compressor, condenser fans, and defrost timer. Apparently the fans can be cleaned fairly easily, and the timer can be replaced if its electric motor starts making noise.

In my case, we called in an independent service technician (found via Service Magic). He diagnosed the problem for us at no charge. Unfortunately for us it was the compressor, and a new unit would cost almost as much as a new fridge, so we ended up getting a new fridge.

But in your case, maybe you're lucky and it's one of the other parts.

Some motors have a start up noise, and is normal for that type of motor. My table saw does this. Starts out a loud 60 Hz hum which quickly increases in frequency as the motor spins up, then gets inaudible or washed out by rushing air when full speed is reached. Motors that do this are not really appropriate for residential blowers.

It also possible it's not a normal noise and there is a defect. Very hard to differentiate on the internet. The rhythmic noise sounds like the squirrel cage is slightly hitting something. Perhaps it is misaligned or out of balance. Misalignment or balance issues could also cause the motor to make unusual noises. Something is not right and warrants further investigation. Could very well be minor, or a warning of something that will get worse.

Even if the start up noise were normal, I would be unhappy that a formerly quiet motor was replaced with a noisy one.