Materials like this are actually quite good at not trapping humidity but allowing it to escape and evaporate. Turf and clay are actually making a comeback as an effective way to insulate houses for this reason. Turf and dry soil actually are a bit hydrophobic. They don't absorb water that well unless you completely drench them. That's actually a problem in dry regions. Because rainfall just erodes the soil instead of absorbing into it.

Having grass growing on the outside probably improves things. What little moisture is in there gets used by the grass. And the roots hold the whole thing together. And any excess water just runs off.

Mold is what happens when moisture can't get away. It's a problem in modern construction when using materials that don't let the water through. Like concrete.