They showed the interior of the earth with other megafauna, but how exactly Godzilla or Kong are getting their caloric intake satisfied on a regular basis is somewhat of a question regardless.
Godzilla especially... feeds on radiation? But not just like, consuming uranium ore. He can take a full thermonuclear blast to the face and seemingly heals bodily injury. Maybe makes him feel really full too?
It's handwaved at best, which is fine. Trying to figure out how Kaiju work is like trying to explain The Force with physics. It's just magic, don't worry about it.
There are some real life fungi that are radiotrophic (like photosynthesis but with radiation [ok light is also a kind of radiation but you know what I mean]) So at least part of that makes sense but not the adsorbing a nuclear blast bit.