When a starship is destroyed in a battle with another starship in deep space the resulting anti-matter warp core explosion can be detected by anyone within 5,000 Light-Years. The explosion point will be plotted as a purple "X" (4 spokes) on the starmap. You will also receive a distress call from the ship with will give you the name and location of the destroyed ship; this will appear in the Reports screen.
You are unable to determine the ID number or race that the ship belonged to. Ships using their Glory Device do not send a distress call, but their victims will if they are destroyed.