What could it mean when a 12 year old fridge leaks on the inside.

There is water in the crisper trays, and food at the back of the fridge on the shelves is frozen.

I cannot hear the fridge making any noise such as a compressor, fan etc.

The back of the fridge has a metal plate covering the insides.

If I take the back off the fridge, what should I look for to determine what the problem might be, and does it sound like I might be able to fix it myself possibly?

Fridge. No freezer, no icemaker?
Confusing as to water being collected. Sounds like fridge door being left open.

Remove the rear panel. You may see the condenser cooling coil
Totally frosted and iced up. Little air flow can get through that to properly cool the fridge.

Ok, I will see what I can see with the back off.

Weird it hasn't made any noise. Keeping fingers crossed this isn't going to be a costly repair.

Thanks for your help.

If it is iced up and froze up you can fast defrost it with a wet vac blowing in the coil and another sucking out the melt water.
It will work normal a day or so while you repair the problem.