What's burning and can I fix it?

My Kenmore fridge - model 25379132990 (taken from sticker inside of fridge) with top freezer w/icemaker was making a rubbery burning smell. Pulled it from the wall to see light smoke coming from the compressor area - there is some kind of box with wires attached and that's where the smoke was coming from.

Would replacing that part fix it? Everything else seemed to be working fine - except you could hear the compressor attempt to come on and then shut off. What is that part called? The fridge is 5 years old and was cooling fine up until this point.

Thanks!


RC