My fridge is making this weird loud humming noise and I want to fix it without messing with the internals. Any help?

Need to figure out where it is coming from first, if the compressor. As others said, then clean the condenser coils and ensure there is enough air flow around it. If it sounds like it is coming from inside, it could be the evap fan, could be dying, ice build up and such. Could also be the condenser fan, but you can tell if that is working just by looking and such.

Probably you need to level it

Sometimes either pushing it tighter against the wall or pulling it slightly away from the wall will do it. The motor vibrates and might be just touching the wall, which can make more noise.

Grab the door one each side and simply throttle the entire refrigerator. I have seen this work.

Sometimes just a smart kick can do the trick. Yes, I'm serious.

If that doesn't work, THEN do all the stuff the other answers suggest.

Check if the fridge is level and level it if it is not. Unplug it and completely defrost it for 2 days. Clean the space in the back (where the compressor is located) with a blower or a vacuum. Check if the humming goes away.

No joy, it is time for a new fridge it the humming annoys you. The compressor could be on its last legs.

Unplug it, open the doors and let it sit for a day. While it is down, clean it up inside and out, front and back. Plug it back in. If that doesn't fix the problem, go buy a new one to replace this one so that your life can return to normal. Then, take this one and see if you can figure out how to buy a part to replace. If you get it working, you will have 2 refrigerators which can be handy and avoid the panic crises when one has a problem.

Clean all the dust from under the machine. Pay close attention to the fan blades and clean them if possible. Also make sure the fridge is leveled. When leveling a refrigerator, you're NOT looking for the unit to be perfectly level. In fact, it should sit back a bit, with the front edge of the refrigerator higher than the back.