To me, destruction of the moon doesn't necessarily mean it exploded, but rather its relationship with the earth was destroyed. Such as in the remake of The Time Machine, where part of the moon broke off while engineers were preparing it for colonization. This would destabilize its orbit causing a major change to the tidal stresses it applies on the earth. The ocean tides would change, there would be more tectonic activity, etc. Yes, some of the debris would strike the earth or burn up in the atmosphere, but most of it would remain in orbit or fly off into deep space.
Unlike how the future moon was depicted in The Time Machine, the pieces wouldn't stay as close together. The force that caused the split would have changed the velocities of the major pieces, thus changing their orbits. The smaller section that breaks off would most likely have been accelerated, thus raising its orbit. The same force, would have slowed the larger section, thus lowering its orbit. The orbits would be more elliptical than they are now, making the tides much less predictable.
Why would we be mining on the moon? Either we found much needed resources there or more likely preparing under ground areas to build habitation in, so they would be better protected from asteroid and meteor impacts. The reason we would be doing this is the earth is overpopulated and we were using the moon as a first step to colonizing the other planets and moons in our own solar system before striking out to other solar systems.
