Let's say a Comet fmashed into it, or an alien race with just enough technology to fly across the galaxy drilled a whole in the Darkside and dropped in a Mega Atomic bomb.
I can imagine that we would still have reflected light at night for many years, but eventually that would dissipate over 100's or even thousands of years.
What else would happen to our world?
What impact would this have on our Galaxy?
From your second sentence I assume by "explode" you mean cracked apart (rather than the more traditional star wars death star type explosion)
If so then you first question you partially answered, if the Moon remained but broken into pieces as in your image we would still have reflected light from and would be able to slowly watch it break apart. Any explosion powerful enough to crack open the Moon would surely send at least some particles hurtling in many directions, including toward Earth, so we would probably have a big meteor shower for a few days after the event.
As to what would happen in the long term it would depend on the power of the explosion. The Moon might crack open, and be further destabalised by the tidal forces Earth puts on it so slowly breaks apart... but unless the explosion is big enough to send large percentages of it in opposite directions the Moon would eventually reform, though the surface would probably be a lot smoother. It could also have destabalised the orbit so the Moon may slowly be pulled in towards Earth, or more likely (given that it is already happening a little bit) be carried away from Earth faster. It also might start a rotation so we don't see the same face all the time.
Any explosion powerful enough to break the Moon apart for good would not take 100s of years.
Here is an alternate question for you... What if the Moon were a disco ball?