Ok my answer is HEAVILY spoiler filled so be warned if you haven't seen it.
Thay being said there are several parts to this answer. The first is that this species has, according to the interviews I've seen, been around as long as Godzilla. The MUTO can multiply very quickly and cause a lot of damage to the earth and the Godzilla's evolved as natures way to keep the MUTOs in check. That's part of it, it's his instinct.
Now, if you look at the part with the skeleton, that was not a Muto skeleton- the spinal column was all wrong for that creature- it was the skeleton of an ancient Godzilla. Where was the Muto's eggs laid? In that old Godzilla. This entails that once the earth became less radioactive and the creatures, Godzilla and Mutos alike, were driven underground, the Mutos had to find different sources of radiation to feed their young. My guess is that the Mutos started fighting and killing their alpha predators, Godzilla, and using the carcasses of the Godzilla's as a location to hatch eggs, at least male ones, which would explain why the egg sacs were there. This would further explain why Godzilla was so intent on fighting and killing the Mutos- self preservation. Should the next have hatched in SF in the movie, Godzillla would have been vastly outnumbered by a species that just two of almost killed him.
Anyways that's my theory