Galaxies as such do not explode. What you have is an Active galactic nucleus, gas and dust and entire stars falling into a massive black hole.
The black hole doesn't explode either: nothing can get out of it (apart from very small black holes perhaps evaporating due to "Hawking Radiation"). It's matter getting heated as it falls very fast but still outside the Event Horizon.
Galaxies are fairly stable, though larger galaxies may shred and absorb them. Or two may merge into a new and larger galaxy. Our own Milky Way and the Andromeda Galaxy are heading for such a merger.
I hope this helps!
