Well, a "space explosion" like a supernova for example isn't so much a traditional explosion in the sense of "boom-stuff, oxygen and a flame" like we make things go boom over here on Earth. Most "space explosions" are because an object got so heavy that it collapses into itself, creating a fat boom because it's a lot of matter that can't sustain itself anymore. Technically, they're huge nukes.
The aftermath of such a big boom are huge clouds of "space dust". This "space dust" contains a huge range of materials and, over time, will create new planets and stars and solar systems.
In a nutshell, anyway.