Here's something strange. Not all wall outlets in a typical home isn't GFI protected. Typically, kitchen, bathroom, and garage outlets are protected, but no where else. So please re-check your findings.
Assuming it IS really tripping the GFI that is installed on the wall and looks like an outlet...
What that means is, you likely have some "leakage" somewhere in your enlarger. You have two wires in A/C outlet. One is the supply line (black) and the other is the return line (white). All of the current from supply should return to return. Well, in your case, it doesn't.... some of it leaks to the chassis and your GFI breaker is sensing this. If it weren't for the 3 prong plug where one is ground, you may even feel a shock when you touch metal portion of your enlarger.
You are dealing with fairly old unit. Your transformer could have lost insulation. Some rodent could have chewed on wires. Maybe there's heavy dust on something which in turn got wet creating a pass for electrical power to leak.
As to what you can do, really depends on your skill level. What I suggest you do is to unplug from the wall. Let it sit for 30 minutes or so to let any charges bleed through. Then start your visual inspection. Start from your plug. Then the wire, all the way to the chassis. Any cracks, loss of insulation, etc, should be noted and repaired. Then, carefully open the cabinet and see if you see heavy accumulation of dust, loss of insulation, or any damage that are visible. Old transformers can leak and that damage may or may not be visible as it is an internal damage in the windings and insulation. If there is a plug, note which direction it plugs in and unplug it, then clean, then plug it back in. Do this one at a time.
Since you have no electrical experience, we are strictly looking for damages that you can see.
If you plug this into an outlet without a GFI, it will probably do something but I DO NOT RECOMMEND THIS. If the damage is substantial, you may be risking your safety. If you should do this CAREFULLY JUST TO TEST, and you find regular circuit breaker trips too, then you have a dead short somewhere and that's a whole different problem.
I'm going to stop here and see what you can find. Please make sure it is unplugged before you attempt any of this, and proceed slowly and carefully. Take notes on how you disassembled and unplugged anything.