According to ja.Wikipedia:
Magic Stones are the core of the monsters' life force, and adventurers' means of living. Without exception, it locates in the monster's chest.
Killing the monsters, gouging the Magic Stones out of their chests (in the anime, the body of the monsters disappear when defeated, leaving their Magic Stones behind), and cashing them in at the guild is the main source of income of the adventurers.
Charged with magic, Magic Stones can be used to create various tools, such as lighting, sewage purification, cooking stove, refrigerator, etc.
Monsters die an instant death if the Magic Stone in their chest is destroyed by a direct attack.
DanMachi Wiki has a bit more details regarding the use of Magic Stones:
- Magic Stone Lamp: A handy and flexible use of the Magic Stone is the production of Magic Stone Lamps. These items give off a dim shine that allows one to see underground or in total darkness. It was once considered as the "century's greatest invention."
- Ignition Device: Magic Stones can be used to produce ignition devices.
- Freezer: Magic Stones are used in the production of freezers to preserve food.
Volume 1, Chapter 2 is cited as the source of the information above.