Requirements
Anyone can find a shooting star, and will be rewarded if they are the first to do so, but you will need a Mining level of at least 10 to mine the star itself. To mine the thickest, hardest layers, however, a higher Mining level will be extraordinarily useful.

Rewards
Star sprites, being magical creatures, are capable of using the stardust you gather from their homes to create objects and items, and are more than happy to do so. To claim your reward, simply chat to the star sprite while it stands around in the remains of its star. Note that you can only claim a reward once per day, so you might like to hold onto your stardust until you have some more (up to the maximum of 200).
Regardless of this, there is also a chance that gems can be found in the various layers of the star. The harder layers have been subject to the greatest heat and pressure, so you are much more likely to find one in the largest stars with the hardest shells.