Finger independence is basically the ability to control each finger. It's the key to a great, relaxed technique. It's having the mechanics and coordination to play whatever you want and how you want. Also, benefits to finger independence include injury and playing faster.
When should you study this? When you're prepping for advanced music or already playing, when your note and rhythm accuracy is on point but your playing is lacking, or when your hands and fingers are stiff ("sticky" fingers).
The hard thing is how boring this one simple exercise can be. So make sure to practice in a variety of ways. You can play in all the major and minor keys, hold down more than 1 finger at a time, play with different rhythms, play scales and arpeggios, practice with different rhythms, use the actual music you're practicing, and lastly you can practice without a piano.
If you attempt this make sure to use extreme caution. Start small, do very short sessions. Be careful not to injure yourself (it can happen). And remember to be patient, this can potentially take years to master.